{ "cells": [ {"cell_type":"markdown","source":"

Vectors and Vector-Valued Functions

","metadata":{"internals":{"slide_type":"subslide","slide_helper":"subslide_end"},"slideshow":{"slide_type":"slide"},"slide_helper":"slide_end"}}, {"cell_type":"markdown","source":"

In these notes, we will need these packages. The first is in the standard library, the others must previously have been installed.

","metadata":{}}, {"outputs":[],"cell_type":"code","source":["using LinearAlgebra # for norm, dot, and cross\nusing Plots # for plotting\nusing SymPy # for symbolic math\nusing Roots # to find zeros numerically\nusing QuadGK # to integerate numerically"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

(If the MTH229 package is installed, the last four lines could simply beusing MTH229`.)

","metadata":{}}, {"cell_type":"markdown","source":"

Vectors

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In julia, creating vectors is very straightforward: in place of the \"book\" notation $\\langle 1,2,3 \\rangle$, we use square brackets, as in

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["3-element Array{Int64,1}:\n 1\n 2\n 3"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["[1,2,3]"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

In Julia, this creates a vector. The output describes itself as Array{Int64,1}. This translates into an array of integers with one dimension. An array is a collection of objects stored in a multi-dimensional grid. For vectors the dimension is 1, for matrices – two-dimensional, rectangular arrays – the dimension is 2. This is important, note the difference if we leave the commas out of the above:

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["1×3 Array{Int64,2}:\n 1 2 3"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["[1 2 3]"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

This creates a 1x3 Array{Int64,2}, which has 1 row, 3 columns, 2 dimensions and stores integers. The only moral here – use commas to create vectors.

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The [] notation is used in many different ways in julia. In the above we see its use to combine like-type items into a collection, in this case a vector or matrix. As well, the [] notation is used to access vector components, for example x[2] would be the second component of the collection x. It is also common with list comprehensions of the form [x^2 for x in 1:5], though this is related to the first use. More generally, identifying the different uses of matched braces: [], (), and {} is important in trying to read julia syntax.

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","metadata":{}}, {"cell_type":"markdown","source":"

Algebra of vectors

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Vectors have many arithmetic operations defined for them. These fit naturally into julia's syntax.

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["([2, 4, 6], [5, 7, 9], [3, 2, 1])"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["u = [1,2,3]\nv = [4,5,6]\nw = [1,3,5]\n2*u, u + v, v - w"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

The number of entries in a vector is determined by length:

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["3"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["length(u)"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

The norm of the vector is returned by norm, but first you need to load the LinearAlgebra package, as was done above. The default is to use the Euclidean norm (p=2), though others are possible:

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["3.7416573867739413"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["norm(u)"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

This makes finding a unit vector trivial:

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["3-element Array{Float64,1}:\n 0.2672612419124244\n 0.5345224838248488\n 0.8017837257372732"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["u/norm(u)"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

We make a function to compute the unit vector of any vector:

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["uvec (generic function with 1 method)"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["uvec(u) = u/norm(u)"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

If we give our variable names u\\hat<tab> then a \"hat\" will match the book for a unit vector:

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["3-element Array{Float64,1}:\n 0.2672612419124244\n 0.5345224838248488\n 0.8017837257372732"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["û = uvec(u)"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

(There is nothing but a naming convention. Using a hat does not by itself make something a unit vector.)

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Here we verify that the triangle inequality holds for u and v:

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["true"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["norm(u+v) <= norm(u) + norm(v)"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

The dot product is defined for any two vectors of the same length and is implemented via dot:

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["32"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["dot(u, v)"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

For those who like to match the text book, you can use \\cdot<tab> as an infix operation:

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["32"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["u ⋅ v"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

using $\\hat{u} \\cdot \\hat{v} = \\cos(\\theta)$ To find the angle between the two vectors u and v we have:

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["0.2257261285527342"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["acos(uvec(u) ⋅ uvec(v))"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

The projection of u along v is given by the projection formula $u \\cdot \\hat{v} \\hat{v}$:

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["3-element Array{Float64,1}:\n 1.6623376623376622\n 2.0779220779220777\n 2.493506493506493 "]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["ev = uvec(v)\n(u ⋅ ev) * ev # parentheses are not needed here, as dot happens before *"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

Lets break u up into a parallel and perpendicular components in terms of v:

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["3-element Array{Float64,1}:\n -0.6623376623376622 \n -0.07792207792207773\n 0.506493506493507 "]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["ev = uvec(v)\nupar = (u ⋅ ev) * ev\nuperp = u - upar"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

And we check that uperp is indeed perpendicular, as it should have a 0 dot product with v:

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["4.440892098500626e-15"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["uperp ⋅ v # 0 up to roundoff errors"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"
","metadata":{}}, {"cell_type":"markdown","source":"

For 3D vectors, like u and v, the cross product is defined and implemented in cross:

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["3-element Array{Int64,1}:\n -3\n 6\n -3"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["cross(u, v)"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

Again, an infix operator is available: \\times<tab>:

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["3-element Array{Int64,1}:\n -3\n 6\n -3"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["u × v"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

Here we verify for these vectors that the cross product is not commutative, but rather anti-commutative:

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["([-6, 12, -6], [0, 0, 0])"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["(u × v) - (v × u), (u × v) + (v × u)"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

Here we verify that the cross product is not-associative

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["3-element Array{Int64,1}:\n -17\n -2\n 13"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["left = (u × v) × w # also cross(cross(u, v), w)\nright = u × (v × w) # and cross(u, cross(v, w))\nleft - right"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

And this finds, again, the angle between u and v:

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["0.22572612855273397"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["û, v̂ = uvec(u), uvec(v) # u\\hat and v\\hat to enter\nasin(norm( û × v̂ ))"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

Let's verify that $\\| u \\times v \\|^2 = \\| u \\|^2 \\| v\\|^2 - (u \\cdot v)^2$:

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["-2.2737367544323206e-13"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["left = norm( u × v )^2\nright = norm(u)^2 * norm(v)^2 - (u ⋅ v)^2\nleft - right # up to round off error"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

Vector-valued functions

","metadata":{"internals":{"slide_type":"subslide","slide_helper":"subslide_end"},"slideshow":{"slide_type":"slide"},"slide_helper":"slide_end"}}, {"cell_type":"markdown","source":"

A general definition of a function is a mapping from a domain to a range. A vector-valued function is a function which takes a value from the real line and returns a vector. In shorthand, $f: R \\rightarrow R^n$, where $n=2,3, ...$. In this section, we see how to define such functions, how to operate on them, and visualize them. In this specific case of a function from the real line to 2 or 3 dimensions, the term parameterized curve is used to describe the function.

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Two possible approaches

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In julia two reasonable approaches to implementing a vector-valued function are:

","metadata":{}}, {"cell_type":"markdown","source":"","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["vv (generic function with 1 method)"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["vv(t) = [sin(t), cos(t), t]"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["3-element Array{Function,1}:\n sin \n cos \n getfield(WeavePynb.ZKWQZSUDT, Symbol(\"##1#2\"))()"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["vf = [sin, cos, t -> t]"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

The latter uses an anonymous function in its last component. One can convert between the two representations using a comprehension:

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["as_vv (generic function with 1 method)"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["as_vf(vv,n=length(vv(0))) = [t -> vv(t)[i] for i in 1:n]\nas_vv(vf) = t -> [f(t) for f in vf]"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

(The as_vf function needs to know the $n$ in $R^n$. It can be specifed as a second argument or, by default, is computed from the value of vv(0).)

","metadata":{}}, {"cell_type":"markdown","source":"

The function that returns a vector is generally easier to work with, except for the case of plotting, so we will functions that return a vector (vv).

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Some examples

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Parameterizing a line

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As with most cases in math, we begin with lines. A line in 2 or 3 dimensions can be thought of in terms of scalar multiples of a vector from a fixed point. Let $p$ be the fixed point, and $v$ the vector, then we can represent any point on a line going through $p$ in the direction of $v$ by:

","metadata":{}}, {"cell_type":"markdown","source":"\n$$\np + t v, \\quad -\\infty < t < \\infty.\n$$\n","metadata":{}}, {"cell_type":"markdown","source":"

In julia, we have, for example:

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["([1, 2, 3], [4, 4, 4], [7, 6, 5])"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["p = [1,2,3]\nv = [3,2,1]\nf(t) = p + t*v\nf(0), f(1), f(2)"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

A helix

","metadata":{"internals":{"slide_type":"subslide"},"slideshow":{"slide_type":"subslide"},"slide_helper":"slide_end"}}, {"cell_type":"markdown","source":"

A typical helix is parameterized by

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["helix (generic function with 1 method)"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["helix(t) = [sin(t), cos(t), t]"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

A pringle

","metadata":{"internals":{"slide_type":"subslide"},"slideshow":{"slide_type":"subslide"},"slide_helper":"slide_end"}}, {"cell_type":"markdown","source":"

Another example we will use, is this function, which when graphed will make a \"pringle\":

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["pringle (generic function with 1 method)"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["pringle(t) = [cos(t), sin(t), sin(2t)]"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

Two-dimensional vector-valued are produced in a similar manner with just two components being satisfied.

","metadata":{}}, {"cell_type":"markdown","source":"

Visualizing vector-valued functions.

","metadata":{"internals":{"slide_type":"subslide"},"slideshow":{"slide_type":"subslide"},"slide_helper":"slide_end"}}, {"cell_type":"markdown","source":"

We have used the Plots package for plotting, primarily as it has many conveniences. In particular, we can plot the function $f$ over $[a,b]$ through the command plot(f, a, b). This is a convenience for the task of:

","metadata":{}}, {"cell_type":"markdown","source":"","metadata":{}}, {"cell_type":"markdown","source":"

For plots in 3 dimensions, however, at times must follow steps like the above, rather than use a more simplified interface.

","metadata":{}}, {"cell_type":"markdown","source":"

To review how to do the three steps above, we have

","metadata":{}}, {"cell_type":"markdown","source":"
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generating $x$ values between $a$ and $b$:

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","metadata":{}}, {"cell_type":"markdown","source":"

We have two basic ways to do this: Either through the notation xs=a:h:b which steps from a towards b by steps of size h. Or xs=range(a, stop=b, length=n) which finds n evenly spaced points from a to b.

","metadata":{}}, {"cell_type":"markdown","source":"
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computing the $y$ values, $f(x)$:

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","metadata":{}}, {"cell_type":"markdown","source":"

If we have a function already defined, then the \"dot\" syntax allows one to easily broadcast over the x values, as in ys = f.(xs).

","metadata":{}}, {"cell_type":"markdown","source":"

If we have a mathematical expression to apply to the x values, we can use a comprenshion, as in ys=[sin(2x) for x in xs].

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plotting the paired points $(x,y)$ and connecting with a line:

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","metadata":{}}, {"cell_type":"markdown","source":"

The call plot(xs,ys) will generate a line plot. The scatter function is used to plot just the points.

","metadata":{}}, {"cell_type":"markdown","source":"
Example
","metadata":{"internals":{"slide_type":"subslide"},"slideshow":{"slide_type":"subslide"},"slide_helper":"slide_end"}}, {"cell_type":"markdown","source":"

We make a plot of the sine curve over $[0,2\\pi]$ and its tangent line at $c=\\pi/4$, as follows:

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First, we plot the sine curve:

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":"Plot(...)","image/png":"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"},"metadata":{"image/png":{"height":480,"width":600}},"execution_count":1}],"cell_type":"code","source":["f(x) = sin(x)\nxs = range(0, stop=2pi, length=100) # 100 points from 0 to 2pi, evenly spaced\nys = sin.(xs)\nplot(xs, ys)"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

We add to the graphic, using the related plot! function. Here we add a tangent line at $c=\\pi/4$:

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":"Plot(...)","image/png":"iVBORw0KGgoAAAANSUhEUgAAAlgAAAGQCAIAAAD9V4nPAAAABmJLR0QA/wD/AP+gvaeTAAAgAElEQVR4nO3dd1wUZ+IG8HfKNtilN0XFEhGxgiVi1IhRY8OWoon9jPklas6Yiyma5O4SL4m5Sy4mZ4qn4MXYogkWUMRKVOy9gBgbiggW2Abs7sy8vz8wTY0C7u7s7jzfj3/4WWD2GXaZZ9+ZeWcYSikBAABQKlbuAAAAAHJCEQIAgKKhCAEAQNFQhAAAoGgoQgAAUDQUIQAAKBqKEAAAFA1FCAAAioYiBAAARUMRAgCAorm1CG02W35+fs2/XxAEpV0BjlIqCILcKdwNL7RCiKIoSZLcKdwKL7RXcGsRnjt3btiwYTX//vLycqW9h+x2u8lkkjuFuxmNRofDIXcKtxIEoby8XO4U7mYymex2u9wp3EoUxbKyMrlTuJvZbLbZbHKnqAXsGgUAAEVDEQIAgKKhCAEAQNFQhAAAoGgoQgAAUDQUIQAAKBqKEAAAFA1FCAAAioYiBAAAjyPcuOq250IRAgCAB3GUFN5Ifff6vNep3U2Xp+Hd8zQAAAD3JpZfN2UvrTyWa0geHjLmDUalds/zoggBAEBmktVk3rzCum+TvuvAqLcWslp/dz47ihAAAGRD7VXm7emWnHS/hB6Rr3/NBQS7PwOKEAAAZEBFoWJvtmnjEnWT+Ijpn/Jh9eVKgiIEAAD3orTi0DbT+sV8ZIOw599TRTeVNw6KEAAA3Kfq1H5jRiqj0QU/+4qmWRu54xCCIgQAAPewnz9lXJcqVZgDBo7XtUmSO86vUIQAAOBajqsXTRlp9ivnAvuN8ev4GGE9awo7ihAAAFxFLL9myl52a2rguJlumxpYKyhCAADv069fv5KSErlT3J0kSQzDMJRKlVZqr2S0fqzWn2R+QMgHNfnxFi1aLF++3NUhfwtFCADgfU6cOLFw4cKIiAi5gzhZQUHB+++/7+YnRRECAHilVq1aNWjQQO4UTsbKcfjQs45YAgAAuBmKEAAAFA1FCAAAioYiBAAARUMRAgCAoqEIAQBA0VCEAACgaChCAABQNBQhAAAoGooQAABcSBTFuLg4uVPcC4oQAABcZe7cuV27dj19+rTcQe4FRQgAAE4wadKkf//739X/nzhx4ieffEIIadu27dtvvy1rrvvDRbcBAHzB5iL6n1OSe56rTzQzJf72cdQTTzzx/vvvT58+3WazrVmz5r333iOEJCcnuyfSg0ARAgD4guaBZHws457naqy/yxP16tVr9OjRJSUl+/btS0hIqF+/vnvCPDgUIQCAL4jRMzF36ye3UavVAwcOXLt27fbt20ePHi1jktrCMUIAAHCOJ598csmSJZs2bRo2bJjcWWoBRQgAAM7Rp0+fgwcP9urVKyAgQO4stYAiBAAA59BqtfHx8XfuF6WUypKnhnCMEAAAnMDhcBw/fvzSpUt9+/aVO0vtYEQIAABOsG7duv79+8+bN0+tVsudpXacX4QnTpzw9/d3+mIBAMCTDR8+vKSkxLtOk6nm5CI0Go3jx4+vqKhw7mIBAABcxJlFSCkdP378G2+84cRlAgAAuJQzT5aZM2dOs2bNnnzyyXt8z9WrVxs0aHDn41OmTJk4ceJtD968edNms6lUKieG9HB2u72iokIQBLmDuFVZWVlVVZWiXmiHw2E2myXJTRfE8hDl5eU6nU6j0cgdxH0EQTAaja44Z9LDz8N8EKIolpaW1vanAgMD6/zWcloRbtu2LSsra9OmTff+tvDw8KysrDsfDwoKCgwMvO1BSmlgYKCito82m02j0YSEhMgdxN0MBoPXHWB/EA6Hg+f50NBQuYO4FcuyOp1Oq9XKHcR9BEFgWdYVLzTDyHkRmZpbs2bNrFmzLl++3LZt2wULFsTGxt73RziOq8NvjGXrvoPTaUW4ZcuWnJycX7ZlDMPs2LGjW7dut30bx3FNmzat4TK5nzkrpOdT4CoTRa61JElKW2WiyBeaUqq0Vf6twsLC0aNHZ2dnt2/fft68eRMmTNi1a1dNftDNvzGnHSOcPXs2/RkhhFJ6ZwsCAIBTUFGUO8Lt7rwN07lz50aOHJmUlKTT6caNG+exdyXEhHoAAG9iLywwrkulDtvtj1/Iqzi03T0Z1I1b+iX2vO3BO2/DVL9+/Z49exJCRFF85513RowY4Z54teWSIvTho7gAAHJxlBSaMv9nLywI6Dea1X5721cZrT8f5qY7H3H6oDsf/KPbMG3evPm1117r27fv7Nmz3ROvtjAiBADwdGL5dVP20spjuYbk4SFjXmdUdzmzTBXVSBXVyP3ZfnHnbZgopTNnzty1a9fy5ctrcpqMXFCEAACeS7KazJuWW/dv1ncdGPXWQlbr0dftevLJJz/++ONTp059+eWXhJDc3Nz09PQ9e/bwPG+xWAgher1e7ox3gSIEAPBE1FZpzlltyUn3S3g08vWvuYBguRPdX58+fZ599tmBAwdW34Zp+/btp0+fDg7+NblnHjjDRbcBADwLFQVr7vqr7z/nuHIuYvqnQU9O8YoWJHfchmnWrFn09+SN90cwIgQA8BiUVhzaZlq/mI9sEPb8e6roms669gTeexsmFCEAgEeoOrXfmJnGqLXBz76iadZG7ji1tm7duhdffPGrr77yuqtEoQgBAGRmv5hvzEgTTWWB/cfo2nUjXnL5tNsMHz58+PDhcqeoCxQhAIBsHFcvmjLS7FfOBfQb49/xMfIAF8yEOkMRAgDIQLxZasxabMs7YOj9dMj4WQyvoLsLeBoUIQCAW0lWk3nrKuueLP8u/SJnLfDwqYFKgCIEAHATaqs0b/vesmOtX4fkqDfns3e7UBm4H4oQAMDlqChU7M02bVyibhIfMX0uH1ZP7kTwKxQhAIArUVqxf7Npw2K+XpOwF/6hqtdY7kBwOxQhAICrVJ3ca89II4agkDGvqZu0cu7C58+fHxgY6Nxlyq64uNj9T4oiBABwPvuFfGPGQtFczncfFt5joNOXv2zZstWrV1utVqcv+cHZbDaO43i+jv0yduxY5+a5LxQhAIAzOa6cN2akCSWFAf3Hqtt1Lysvd8WzdO/evXv37q5Y8oMrLy/XaDQ6nU7uIDWFIgQAcA7hxlXThm9sBYcNvUf6T3yH4XhBEOQOBfeHIgQAeFC/mxo4cyGr9ZM7EdQCihAAoO6kqgrLtlWWnRl+nXpHzVzA+gfInQhqDUUIAFAX1GG37MqwbFmpje8U+ep/uOAIuRNBHaEIAQBqidLKozuN6xZyofXCXviHd901EO6EIgQAqDFKK4/vNmYu4gxBIWPeUDeOkzsQOAGKEACgRuznTxnXpUoV5sD+Y3TtPXTqAtQBihAA4D4cl88aM1KF68UBA8b6JTzqpTfOhT+CIgQA+EPC9Sum9f+znT1h6PuMf5d+DIdtpg/CiwoAcBeSxWje9n311MComS8zGq+5TgrUFooQAOB3pEqLectK6+4N/g/3jZq1kPUzyJ0IXAtFCABwC3XYLTvWmreu0rXuEjnjCy4oTO5E4A4oQgCAn6cGrl3AhdUPn/Ih7hqoKChCAFA2SiuP7TRm/o8LCgsZP0vdKFbuQOBuKEIAUC5bwWFjRhqhJOiJydoWiXLHAXmgCAFAieyXCkxZS4TiC4beI/yT+mNqoJKhCAFAWYTSy8bM/9kv5gX0fdZ/4juE5eROBDJDEQKAUojGG6aNSyqP7TIkPxEyegajUsudCDwCihAAfJ9UYTZv+c66Z6N/Uv+oWQtZnV7uROBBUIQA4Muo3Wb5cbV5+w+6to9EvvYlFxgqdyLwOChCAPBNVBQq9mabNi5RN4mPmPYJHx4tdyLwUChCAPA5lFYc+dG0/hs+NCps0t9VDR6SOxB4NBQhAPiUqvyDxow0huWCn35J07y93HHAC6AIAcBH2AtPG9eliaYbgf3H6tp1w9RAqCEUIQB4PUdJoSlzkb3wTED/0f6d+hCWlTsReBMUIQB4MbH8mmnDt5Un9xoeeypkzBuYGgh1gCIEAK8kVZjNW1beunHurAWYGgh1hiIEAC9D7VXmnHTL9nS/hB6Rr3/NBQTLnQi8G4oQALzG76YGvvxvTA0Ep0ARAoA3qL5xbkYaFxKJqYHgXChCAPB0toLD5WsXMhwfPGKapnk7ueOAr0ERAoDnsl/MN2akiaaywP5jMDUQXARFCACeyFF8wZS5yH7lXGC/MX4dH8PUQHAdFCEAeBbhZolpw2Jb/kFD76dDxs9ieJXcicDHoQgBwFNIVpN566rqqYGRsxawWn+5E4EioAgBQH7UVmne9r1lx1q/DslRb85n9UFyJwIFQRECgJyo4LDuyjRvWaFp0SHiL5/zIZFyJwLFcfLx56ysrPj4+KCgoPj4+OzsbOcuHAB8CqWVR3aUfPh85cm9Yf83O2TUq2hBkIUzR4SSJI0aNeq7775LTk7+4YcfJkyYUFRU5MTlA4DPqDyxx5S5iNX5h4x6Vd2kldxxQNGcWYSCICxevLhXr14Wi0Wj0QQF3WUvv8lkevvtt+98vEePHklJSbc9aLVaOY5TqRR0zpjNZquoqFCrlXUFfavVyjCMotba4XBYrVatVit3ELeyWq2iKNrO51VmL6ZWk7bXCFXrJDshdotF7miuIgiC1WrV6XRyB3Erq9XqcDhEUXTnk2q1Wp6vY6M5swjVavWAAQMsFktAQADDMDt37rzrt9nt9jsfFASBUnrbg/RnTgzp4RS4ykSRa63AVSaE0NJL9tw19htXtI+NVLXtRhjG538DCn2hvW2tnX+yjF6vt1gsc+fOnTZt2v79+2/7akBAwJw5c2q4KJvNZjAYFDUiVKvVLMsaDAa5g7iV3W43GAxKGxFKkqScF1q4UWxav5g9fVCX/FRQz6EMp5TT9ARBEARBOS90NVEUNRqNF42DnXmyzIULF2bMmEEI8ff3nzhxYl5enhMXDgDeSLKajOtSSz+ZxgWF6ab+W/fIIOW0IHgLZxZh/fr1Fy5cmJOTQyldsWJFQkKCExcOAN5FqrKa1v/v6vvPEVGImrkgMOVPjMZrhgigKE4+Rpienv7KK6+cP38+Li4uNTXViQsHAG9BHXbLrgzz5u90rTpHvvofLjhC7kQA9+LkfRSPPvrowYMHnbtMAPAa1XcNXJfKhUaFv/i+Krqp3IEA7g876wHAOSqP55oyF7H6wJAxr6sbx8kdB6CmUIQA8KBsPx0zZqRShyNwyCRty05yxwGoHRQhANSd4+pFU9a39sLTAb1H+nfph7sGgjdCEQJAXYg3S02bl1cd363vOSxk9Gu4ayB4LxQhANSOZDGat31ffdfAqLdSMSkCvB2KEABqSqq0mLestO7e4P9w36hZC1k/ZV0wBXwVihAA7o867JYda81bV+lad4mc8QUXFCZ3IgCnQRECwD1VTw1cu4ALqx8+5UNVvcZyBwJwMhQhAPwBSiuP7TJmLuICw0LGz1I3ipU7EIBLoAgB4C5sBUeMGamE0qDhL2rjOsgdB8CFUIQA8DuO4gumjUschQWG3iP8k/oThpE7EYBroQgB4Bah9LJx/f/sF/IC+j4bOvYNwnJyJwJwBxQhABDReMOU9W3l8VxD8pMho2YwKgXdJBkARQigaFKFxbzlu1uz42cuZP30cicCcDcUIYBCUbvN8uNq8/YfdG0fiXztSy4wVO5EAPJAEQIoDhWFir3Zpo1L1E3iI6Z9wodHy50IQE4oQgAlobTicI5p/Td8WL2wSX9XNXhI7kAA8kMRAihFVf5BY0Yqw6mCR0zTNG8ndxwAT4EiBPB99sLTxnVpoulGYP+xunbdMDUQ4LdQhAC+zFFSaMpcZC88E9B/tH+nPrhxLsCdUIQAvkksv2ba8G3lyb2Gx54KGfMGpgYC/BEUIYCvkSrM5i0rb00NnLWA1WFqIMC9oAgBfAe1V5m3p1ty0v0SHo18/WsuIFjuRABeAEUI4AuoKFh3bzBnL9M0bxcxfS4fVk/uRABeA0UI4OWqb5ybkcaFRIY9/y6mBgLUFooQwItVndpvzEhlNLrgZ6ZrmrWROw6AV0IRAngl+/lTxow0yWoKGDhe1yZJ7jgAXgxFCOBlHFcvmjIX2YvOBvYb49fxMUwNBHhAKEIAryHeLDVmLbblHTD0fjpk3EyGV8mdCMAXoAgBvIBkNZm3rqqeGhg5awGr9Zc7EYDvQBECeDRqqzRv+96yY61fh+SoN+ez+iC5EwH4GhQhgIeiomDdmWHevEITlxjxl8/5kEi5EwH4JhQhgOe5NTUwlQuJCnthtiq6mdyBAHwZihDAs1Se2GPKXMTq/ENGvapu0kruOAC+D0UI4Cls504Y16VSW2Vgyp+08Z3ljgOgFChCAPk5rhaashbbL54O6DPSv0s/TA0EcCcUIYCchBvFpvWLbWeOGPqMDBnzOsPhTxLA3fBXByAPajWW53xXcWi7vsfQ4KdfYjQ6uRMBKBSKEMDdpKoK6+YV9tz16s59omYuYP0D5E4EoGgoQgD3oQ67ZVeGZctKdVwH9cT3ApvFyZ0IAFCEAO5RPTVwXSoXGhX2wj9IREOj0Sh3JgAgBEUI4HKUVh7fbcxcxBmCQsa8rm4cRwhxOBxyxwKAW1CEAC5k++mYcV0qFR1BQ5/XtuwodxwAuAsUIYBLOIovmDYusReeDuiNqYEAHg1FCOBkwvUrpvX/s509Yej7DKYGAng+/IkCOI1oumneuLTi6A5Dz+HBI6czaq3ciQDg/lCEAE5AbZWWnevMW1fp2nWLeuNr3DUQwIugCAEeCHXYLTvWmreu0rXuEjnjCy4oTO5EAFA7KEKAupJE675Npqxv1TFxES/9k49sKHcgAKgLFCFAXdgKDpenf81o/UPGvqlpirsGAngxFCFA7djOnTRlpEqV1oDHR+nad5c7DgA8KBQhQE3ZLxWYspYIxRcMvUf4J/UnDCN3IgBwAidP8l2zZk3r1q2DgoJ69OhRUFDg3IUDyEUovXwj7R83Fr6ra/Vw1Fup/l0HoAUBfIYzR4SFhYWjR4/Ozs5u3779vHnzJkyYsGvXLicuH8D9ROMNU9a3lcd3G5KfCBk9g1Gp5U4EAE7mzCI8d+7cyJEjk5KSCCHjxo378MMP7/weURSLiorufNxgMOj1+tselH7mxJAeToGrTDx1raUKi2Xryoq9G/0efjzijfmsn54SQp0U0jNX2dUUuNYKXGUi01qzD3AVQ4ZS6sQo1URRnDp1Ksuy8+bN++3jeXl5SUlJGo3mzh956aWXJk2adNuDN2/eNBgMKpXK6Qk9lt1ur6ioCApS1nTssrIyvV7vQS+0wy4dyBb3ZbEtOrKPDGEMwc5/BofDbDaHhIQ4fcmezGg0arXau24BfJUgCEajMTQ0VO4gbmU0GjUajVbr1isrBQUF1fmt5fyTZTZv3vzaa6/17dt39uzZd361Xr16eXl5NVwUx3GBgYEetH10PZvNZrValbZ95HneYDCo1fLvdaSiULFnoyl7qbppq8Dpn/Lh0S56IofDodVqw8KUNfterVbrdDo3bx/lJQiCWq0ODw+XO4hbaTQajUaj0+nkDlJTzixCSunMmTN37dq1fPny2NhYJy4ZwOWqb5ybuYgLjgh97m/qhs3lDgQAbuLMIszNzU1PT9+zZw/P8xaLhRBy52E/AA9UlX/QmJHGsFzw0y9pmreXOw4AuJUzi3D79u2nT58ODv71gIorDkACOJH9Yr4xI000lQUOHKdr0xWTIgAUyJlFOGvWrFmzZjlxgQCuI5RcMm74xn4hP6DvM/5dHicsJ3ciAJAHriwDiiOWXzNt+Lby5F7DY0+FjMLUQAClQxGCgkgVZvOWldY9Wf5d+kXNWsDqcAwbAFCEoAzUVmnOWW3JSfdLeDTy9a+5AOdPDQQAL4UiBB9HRcG6e4M5e5mmefuIV+byofXkTgQAngVFCL6rempgRhoXEhn2/LuqBg/JHQgAPBGKEHxT1an9xsw0RqUJfma6plkbueMAgOdCEYKvsZ8/ZcxIk6ymgIHjdW2S5I4DAJ4ORQi+w1FSaNqw+Oepgf3IA1yNHgCUA0UIvkAsv2bKXlZ5LNeQPBxTAwGgVlCE4N0kq8m8ddWtqYFvLWS1/nInAgAvgyIEb0XtVZYda81bV+nadcPUQACoMxQheB8qOKy7Ms2bV2jiOkT85XM+JFLuRADgxVCE4FVuTQ1M5UKiwl6YrYpuJncgAPB6KELwGlUn9xgzFrE6/5BRr6qbtJI7DgD4CBQheAHbuRPGdWm0yho4aIK21cNyxwEAn4IiBI/muFpoylpsv3g6oM9ITA0EAFdAEYKHEm4Um9Yvtp05bOg9MmTM6wyH9yoAuAQ2LuBxRHOZOXtZxaHt+h5Dg59+idHo5E4EAL4MRQgehNoqLTvX3Zoa+MbXnAFTAwHA5VCE4BGow27OzbBsWamN7xT56n+44Ai5EwGAUqAIQW6SJBzJubEjXdOwediUOaqoRnIHAgBlQRGCnGwFh8tXzxc5VcDI6fq4RLnjAIASoQhBHrafjhkzUqnDEZgy0RrRRG0wyJ0IABQKRQju5rh60ZT1rb3wdEDvW1MDrTduyB0KAJQLRQjuI1y/Ylr/je3scUOfZzA1EAA8BLZE4A6i6aZ549KKIz8aeg4PHvkyo9bKnQgA4BYUIbjWb6cGRr05n9UHyZ0IAOB3UITgKlQUKvZmm7IWq5u2jnjlMz40Su5EAAB3gSIEF6i+a+DaBVxY/bAXP1DVayx3IACAP4QiBCezFRwuT/+a0foHj35N0xR3DQQAT4ciBKexFRw2rksjhAQOfV7bArPjAcA7oAjBCeyXCozr0sSy0sCB43XtuhGGkTsRAEBNoQjhgQjXioyZi+wX8gL6Puvf5XHCcnInAgCoHRQh1JFovGHauKTy2C5D8hMhz77KqDVyJwIAqAsUIdSaVGExb/nOuifLv0u/qJkLWT+93IkAAOoORQi1QO02y4+rzdt/0LV9JPK1L7nAULkTAQA8KBQh1AgVhYo9G03ZSzVNW0dM+4QPj5Y7EQCAc6AI4X6qZ8dnLuKCI0Kf+5u6YXO5AwEAOBOKEO6lKv+gMSONYbngp/+sad5O7jgAAM6HIoS7s1/MN2akiaaywIHjdG26YmogAPgqFCHcTii5ZNzwjf1CfkDfZzA1EAB8HooQfiWWXzNt+Lby5F7DY0+FjJrBqNRyJwIAcDkUIRBCiGQ1mTevsO7bpO86MOqthazWX+5EAABugiJUOmqvsuxYa972va7tI5Gvf8UFhMidCADArVCEykVFwbp7gzl7maZ5u4iXP+XD6smdCABABihCRaqeGpiRxoVEhj3/rqrBQ3IHAgCQDYpQcapO7TdmpjEqTfAz0zXN2sgdBwBAZihCBbGfP2XMSJOspoCB43VtkuSOAwDgEVCEiuAoKTRtWPzz1MB+hGXlTgQA4ClQhD5OvFlqzFpsyztg6P10yOjXGF4ldyIAAM+CIvRZksVo2rS84sAWfbeUoFkLWa2f3IkAADwRitAH3ZoauHWVrl23yNe/5gKC5U4EAOC5XFWEoii2atUqPz/fRcuHu6KiYN2Vad68XNMiMeIvn/MhkXInAgDwdC4pwrlz5y5duvT06dOuWDjcHaUVB7aYNizm6zUOe/EDVb3GcgcCAPAOLinCtm3bNmvWLCUlxRULhzvZCg6Xr1nAqFTBz76qeQhTAwEAasElRZicnPxHX7p+/Xrfvn3vfHzEiBHDhg277cGysjJRFFUqBZ3oaLPZKioqav79YmG+Y8tyYq9S9RrBNU+wEmK9ebMmPyhRcs3GFlcxVyuZG3bG5GBMDmJyMCYHYxVuv/Ugw5BAFeUYYlBRLUtC1VKEloZqaKiGRmhosJrWYg3vpqyszOFwqNUKutmFw+EwmUyswuaxlJeXV1VVaTQauYO4jyAI5eXlHKese5kZjUaNRlNZWenOJ9Xr9XXehrj7ZBm9Xj958uQ7H4+NjfX3v/2OB1VVVf7+/ooqQp7nCSF3/iruJJRcqti8zFFY4N/rKW2nPveYGuiQyHkLyTeSAiNTYCJnTOSSlSmtIsFqGu3P1PcjwWoaqCZBatLUnwSoiEF1e7FJlJgcjCARs4NUiORMJdlVxlyvotdtzNVKQilprCdNDKSxnjbVk7gg0iaIhtRmW2ez2fz9/ZVWhKIo1uSF9iV2u12n02m1WrmDuI8gCA6HQ2kvtMPh0Gg0Op3OnU9avfGs4886MUdNaLXaoUOH1vCb1Wq1RqNRVBESQgRBuPdHZuFGsWn9YtuZw4Y+z4SNe5Phbn8Rq0Ry9AY9eP3Wv9NG2lDPxAUycUGkR33muZZMjJ5E+TFqJ41GymzkvJmeN9NzZpJnpisL6fGbVK9iWgeTtiFMxzCmayTTwP9eN7ivfqEVVYQsy1avtdxB3ErzM7mDuA/HcXihPR+mT3gT0Vxmzl5WcWi7vsfQ4KdfYjS/fuAyOciOq/THYinnKj12k8YFMh3CmE7hzAst2bYhjMaVO2aCNSRYwySG/a7qLlroiTJy7CZdepa+tFtUs0xSBNM1kukexSSEMuy9ahEAwK1QhN6B2iotO9fdmhr4xtecIZgQIkhkVwldf0naeoWeNtLO4UyPeuxHndnO4YxW7kMSMXomRk8GNqxuPO6sie4upbtL6fx86VoV7VWf7RPN9IlmYvSoRACQmQuLkNIHPYcCCCHUYbfsyrBsWamN7xT56n+44IhrVWT9GWn9JbqpSGpqYAY0ZD5N4jqFO21Xpys0C2CaBTCjHyKEkMtWurmIbiqibx0QQzTMkBgmOYh71CB3RABQKowIPZgkWfdvNmV9q27wUNiUOeVBjRZckFbuFg5ep72j2QENmblJqii3Ho12jgb+zPhYZnwsoYQ7eJ2uviBNO6wz7mOGNBaHNWYfq89wGCUCgBuhCD2UreBw+er5jEanGTHjB6bVisPSgWuO/g3Zqa3Yfg1Y2fd8OgVDSMcwpmMYN71JeSkxZF5h3jkojt1ORzRlRz3Edg5HHwKAO6AIPY50Ma/0fyup6PyP6L8AAB8RSURBVDjb9U/zHInr90m9o+nklmy/PqzOd1+uZgbyahv21TbsGSNdepaO2S5SQkY1YyfEMo1wHBEAXMl3t6xeyHH5rHHtgsrS4m2tx/7V0T3iOjM+lvm8q6pWc/K8XfNA5q+JzF8T2f3X6OKfpMR08eEIZlIcO6ghy3vwQVAA8F4oQo8g3iw1bV5uOrp7XYPB/2j61jMNtJkt2NbBih4JdQpnOoVzczpxK89LHx+XpuySJsQyL7Rk7z0lEQCgtlCEMhNNN29sWGo+vGNJ5JCshP+Oa6k+EGGNDlfWdSjuQceTsc3Zsc3ZvHL6db7U/gehdzT7cmu2SwTqEACcA0UoG2qrvLJ1beX27zMDup5K/mJ8Qsg7EYzNZrNaMe3kLloGMZ924d7rwKUVSGO2i6Fa8nIr9okmrAr7SwHgwaAIZUAd9p82rhF3fJ9t6Hyj3+cTO0VMw/kgNWNQkT+3YqfGsxmF0qcnpDcPSK+2Yf8U68unEQGAq2H74V6SeGJLNt2y5IhfC+uAOeOSYgIVdH1Np2EZMjiGHRzD7i2lHxyVZh92TGvNTY5nA5R1VVoAcA4Uofsc2nvIse7rG4y/+fHXn+7RxqXX/1SIhyOY1X24k2Xsh0elZisck1uy09twQfhsAQC1gSJ0h4N7D1esT7WL1Njz+UG9OnjytdC8UatgZnFP7ryZnX1Yiv3O8edW3LTWrAGjQwCoGRSha508efrKD2l6a6m5x7g+j/dQ4ephLtPEwCzswf1kYt89JD30neOV1tzUVqw/3uAAcD/YTrjKuXOX8lf9L+p6nqPDk4nDUzQq/Krd4aEA5pueXH45+7dDUux3wjuJ7MRYzMQHgHvB1tn5Skpu7F+xOKZwt6PdE7EvvZaowzErd4sLYpb34g7foK/tEz89IX3QiR0agzIEgLtDETpTpdn843crovOy7bH96r+1sE2QXu5EipYQymzqz2+8TF/fJ358XPqoM5eEafgAcAcUoXNIdlvu6tVB+38or9+12ctfDG8QJnciuOXxBkyfaP7bn6QRW8RuUcyczmxDXKQNAH4DRfigqCjkbc4Sti67HNiKm/DxiPgGcieC27EMGducfbIJO+eomJguTI3nZrRl/fDeBwBCCIrwgVBaemDn1TWLfmLD6eB3nu7agsVIw4P58eTvHbiJLdjX9kktVwkfdWZHNMWBQwBAEdZVRd7Bc9+nFVWyeZ2nTOifiAvEeItGemZ5L27HVTpttzg/X5rXlYsLwucXAEVDEdaa/WJ+4Q9p167dWB07dtyQbo8HY1ThfbpHMfuH8l/mST0yhIkt2LcSOMw4BFAs/PXXglBy6VrmN2Vn8r+IGtFj4uNzmuHiJV6MY8jUePapJuyMvWKrVcKnSZhiAaBQKMIaEcuvmTZ8W3Zs75fhw4UBr/zjYR32hfqGSB35pieXU0yn5IppBfQ/XXFOKYDioAjvQ7KazJuWm/Ztzojsv6TD1/9+NKBTODaUvubResyhYfxHR6UO6cLbCdyUeBbnPQEoB/YF/SFqrzJv+e7qB5OOXK18vPnnVY+Ny3kiEC3oq9QseSuB3ZnC/3BB6rpOOHYTt0cGUAqMCO+CioJ19wZz9rLKmHZ/bv2xPTAqsxvXxIAK9H2xgczWgXxagdRng/BcC/adBA53ywLweRgR/h6lFQe3lrw/qeLkvhXd303SvvJkYv2N/Xm0oHIwhPwplj02XHXaSBLThb2lGBoC+DiMCH9lKzhcvnYhw/Gl/aY9W9g6jmWODueidHLHAjlE6siqx7iV56Vhm4VRzdh3O3A6/K0A+Cj8cRNCiP38KWNGmmQ16fqP/9DeOa1AmpvEPo3LjijeU03Y5HrstN1iu3QhtTvXLQo7BgB8kNKL0FFSaNqw2H4hP6DvM/kPPT5+B20RRI4OV0VgIAiEEELCtGRJMrfmojRiq/hMM2Z2R06Lo4YAvkW5gx7xZunNpR9f/8/rmibxoTNTP9Q8PmiTNKs9u+oxDi0ItxkSwx4dzl+ykg7pwoHrOGoI4FOUOCKULEbTpmUVB7bqu6UEzVqYV6kbu16M9iNHhqsiUYHwB8K0ZEUvbsU5adBG4fk49u0ETqXcj5EAPkVZf8q/TA2kDnvk61/r+4359Iw2OVOY3JJd25dDC8J9jWjKHh6mOnyDdlkrnCrH0BDAFyhlREgFh3VXpnnzCk1cYsRfPudDIi9a6Pj1gkjJ3iGYHQG1UM+PrOvLLzgt9cwQ3krgXmqFq9AAeDcFFCGlFQe2mDYs5us1CZv8gapeY0LItz9Jr+wRZ7Tl/tIGF9OCuniuBZtcjxmzXcwslFJ7cNG4QimA1/LxIrQVHC5fs4BRqYKffVXzUBtCSLmdvLhLPHaDZvfn24di4wV11yyA2ZHCf3BE6rBa+CyJw3wbAC/ls0VoO3fCuC6N2ioCB03QxneufjCnmI7LEVMaMQeG8pgfDQ+OY8hbCWy/hszobeKGy/SzJM6Ae3MBeBsfbAPHlfPGjDShpDCg/1i/DsmEYQghDom8c1D85gxd0J3r3xADQXCmjmHMwWH8y7vFxHRhSTLXGVdmB/AqPlWEYlmpadPyquO79T2HhU54i1HdumfgTyb67DYxSsccGc6Ha+XNCL7Jnyf/7c59f14anC1Ma8293hbHngG8ho8UoWguM2cvqzi0Xd9jaNRbqYzm15kQ35yRXt0rvpPITYnHpglc64km7MMRzJjtYvZl6dueOIMGwDt4fRFKVRWWbassOzP8OvWOmrmA9Q/45UsmB3lxp3j0Jt0ygG8Tgk0SuEMDf2bLAP6jY1LH1cLX3bjBMTiDBsDTeXERUofdsivDsmWlNr5T5Kv/4YIjfvvVfdfoM1vFvg2Y/UNwXgy4FcuQN9qxyfWYZ7eJm4roPx/G5UkBPJp3VoQkWfdvMmUtUTd4KGzKHFVUo99+kRLyr2PSx8fFLx/hhjXG53GQx8MRzKFh/Au7xC5rhGW9uJZB2CcB4KG8rwhtBYfLV89nNLqQ0TM0zdrc9tXSSjIuRzA7yL4hfCM9Nj0gp0A1WZbMpRZIPTOF9ztyE1vgYxmAJ/KmIrT9dMy4LpWKjsCUidqWHe/8hi1X6LgccXxz5m+JHI9tDniGP8WySRHMiK3i1iv0q26YaAjgcbyjCB2XzxozUoXrxQEDxvolPFo9NfC3BIn87ZC46Az95lGuV30MBMGztAxi9g7mp+8RO6wWlidziWF4iwJ4EE8vQqnsWtn2VbemBj73N4a/y8fpy1b67DZRxZIDQ/ko3EECPJKOJ19149IvSP03Cq+05qbHyx0IAH7m0TsQpZ+O3vhsOh8cEfV2muGxp+/agpmXaKfVwoCG7Kb+aEHwdMMas7sH899fkJ7aSsrtGBcCeASPLkI2Ji50xpeGPiMZ9V2uB1O9O3TyLnHlY/wb7XAhD/AOTQ1MbgrfIogkbzfkluCOhgDy8+xdoyoN62e461cKLXTkVjFMyxwaxodq3BwL4IHwLPlHImntVzFss/9rbblX2uBTHICcPHpE+EfWFUqd1whPNGHX9OXQguClBtQT9g3hV56XhmSLN21ypwFQMC8rQodEXt0r/nm3tLoP/xd8jgYvF6NndgziYwNJYrqwpxS7SQHk4U1FWGihj2YIBUZycCjfJQIlCL5AxZJ/Pcx93pUdukn45LiEMgRwP68pwoxC+svu0BDsDgXfktKIrd5NOnQTdpMCuJuTi7CsrCwlJSUkJGTw4MFlZWVOWaZDIjP2ilNzxXTsDgXf1UjP/DiIbx5AOqwW9mI3KYAbObkI58yZExMTU1xc3KhRo48++ujBF3i5gvTMFPKN9NAwPgm7Q8GnVe8mnduFHbJJ+PcJ7CYFcBMnT59IT09fs2aNRqOZOnXqkCFDPvjgg9u+obKy8rvvvrvzB+Pi4mJjY2978MerdPIROi2evhwvMVSoqnJuWE9ks9lsNluVElb1N2w2m0qlkiRJ7iDu43A4/uiF7htJfuxHRu+g2y6T+V1pkNp3CtFmszF3XB/RtwmCoMy/aErpA77WFgfR1+bCvCqViuPqeMMzJxdhUVFRTEwMIaR6XHjnN1RUVHzzzTd3Pv7UU081bNjwtgdDiWVBR+6RSKaq0rkxPZfNZqusrNRq73IBAR9WWVnJcZwoinIHcR+Hw1FZWVlZefd3djhL1nUnfz+h6ZLJpz5sSwj2kd9M9fpS6jvVfl+CINzjhfZVlZWVD/i5dtUl/q/HNbl9KgJVNX23cBznKUX4y6cASuldt2uhoaEZGRk1XForUQwMNKhUCrpcv81mU6vVwcHBcgdxK0mSDAaDWq2WO4j7OBwOlmXv/UJ/8ShZfVF6dqduZnvuz6285ry2e9PpdIr6nCcIAiFEaX/RDMNoNBqdri4XvawSybTdYk4xzerPNQ4Jcnq2u3LyX1f9+vUvXbpECCkqKoqOjnbuwgGUZmgMu3sw/+1P0hObRaNd7jQALlZgpF3WCBYH2T+Ubxvivr3oTi7ClJSU1NRUSmlqauqQIUOcu3AABWpiYHam8A38SWK6cOC6gnYqgtIsPyt1zxAmx7NLkt19204n7xp95513Ro0a1bBhw8TExMWLFzt34QDKpGbJ3CTu0XrSwI3C2wnc1Hgf2U0KUK1KJC/vFrcV0+z+fDs3DgR/4eQiDAoKyszMdO4yAYAQMrwx2z6UeXqLmFNMF3TnAhV0RBV8WYGRjtgqtgxiDgzl3TwQ/AU+WgJ4jaYGZlcKH6XDblLwEcvOSt0zhBdbskvdvjv0tzz7NkwA8Hsajnzelet5Xhq4UZjZjvtza1xrCbxSpUCm75Fzd+hvYUQI4H2eaHLrbNLhm8QyXJsUvE1+Oe2yVjA7yIGh8rcgQRECeKmmBmbXYL6RnnRYLey7ht2k4DUW/yT1yBBeaiXD2aF/BLtGAbxV9dmkyfWlwdnCq205XJIePJxVIFNzxb2ldOtAvnWwB71bMSIE8G5DY9h9Q/j0C9KgjcI1ZV3SErzJiTLaebVACNk/1LNakKAIAXxAIz2TM5BvG8Ikpgvbi7GbFDzO1/lSr0zhjfZsWg/O3/N2RHpeIgCoPZ4lH3TikuvTUdvE5+PYtxJYzrM+c4NCldvJpB3iWRPdmcLHBnromxIjQgDf0TeaOTiMzy2RemUKl6wYGoLM9pTSxHShvh/ZPdhzW5CgCAF8TJSOZPXnBzZiO60WVl9U0C0ewaNIlHxwVBq6Sfi0Czs3idPU8f5IboJdowC+hiHktbZsr/rMM1vFH87Tr7pxfvhDBzcqrWKm7GIFIh0Yyjfw99yB4C8wIgTwTR3DmIPDeJGSh9cIJ8qwmxTcZM1FqfsmbddwsnmAd7QgQREC+LAAFVmSzM1oyz62Xvj8pIQyBJeqFMiUXHH6HunbR+xvtJG86HQtFCGAjxvbnM1N4ZeclVI2CqWVcqcBH3X0Ju20RiizkcPD+E4hotxxagdFCOD7mgUwOwfx7UOZhHRhwyWMDMGZJEo+Pi71WS+83o5dmuyVNwjDMXQAReBZMrsj17cBHZcjDmzIfNQZZ9CAExRZ6bgc0SaRfUP4xgbv2Rn6exgRAihIjyjmyDDeaCcdVuOOhvCgVp6XElcLyfXZ7QO9uAUJRoQAShOoJot7ct+dkwZtFF5qxb3RDteggVort5OXcsX912hGX75TuNe/gTAiBFCip5uyB4fyOcVSjwzhjBFDQ6iFzUW03Q9CsIYcGuYLLUhQhACKFe3PbOzPj2zKdl0n/OcUJlfA/VUK5M+7xYk7xIXduc+SfOcwM4oQQLkYQl5qxe5K4Zf+JPVZLxRa0Ibwh/aU0oR04aaNHB3O9472hYHgL1CEAEoXG8jsSOH7RLMdVwupBbg8KdyuSiSv7ROf2Cy+34n9ticX5IUTJO4NRQgAhGPI6+3YLQP4eaekARtx5wr41d5SmpguFFrI0eH88Ma+WRm+uVYAUAdtQpi9g/lHItkO6cL8fBw1VLoqkbyxXxy2WXi3A7u8FxemlTuQy6AIAeBXPEtmtWe3D+JTC6Q+64XzZrShQv14lbb7QThvJkeHq55s4uNN4eOrBwB1EB/E7Erh+zVkH14jzD0hiWhDJTE5yIu7xFHbxH92Zlf04sJ9dyD4CxQhANwFx5BX27C7Uvg1F6WktcLRmyhDRVhXKLVeJUiUnHiCHxyjlILwlWkgAOACzQOZLQP5tAKp7wZhQiz7ToLvTB2D2xRZ6bQ90rGbdHFP7tF6PjU74r6UUvgAUDcMIX+KZY8PVxVaSNsfhOwiDA19jUjJZyel9ulCqyBybDivtBYkGBECQE1E6MjSZC7rMp28S0wMZf7dhY32kpuPw70dvE7/b6cYoCI7U/gWgQp9TTEiBICa6teAOfEE3yqYSUgXPj4uOTD53puV2cjUXHHQRuHPrdgtA5XbggRFCAC1ouXIXxPZ3YP5rVekxHRhezH2lHofiZL/5kvxqxyEkFNPqsY2Z5XbgYQQ7BoFgDpoFsBkPs6nX5Am/Ch2DGP+2Zn16tvRKcqeUvpSrqjjSVZ/vl0IXjVCMCIEgDob1pg99QTfLoTptEZ464BoFeQOBPd02UrH5YhPbRFfbs3mDEIL/gpFCAB1p+PJWwnskWH8RQuJWyks/knCldk8kMVB3jkotv9BaOhP8p7kRz2k9H2ht0ERAsCDivZnFvfkVj7GfZUndVwtbMIUC48hUbLwtNRipXDeTA4P52d35PQquTN5HhwjBADn6BLB7Erhf7ggvZQrxujJnM5c+1AMPOS0rlCadUAKUpPVfTjfuJW8i6AIAcCZhjdmBzdiF5yWBm4UH6vP/DWRbRaATbC75RTTmQdEs4P8oyOb0gh7/u4DvyAAcDKeJS+0ZE8/xTcPZLqsFSbtEC9asLPUTQ5ep/2zhIk7xMkt2SPDeLRgTeB3BAAuoVeRtxPYgqdUUX6kQ7owJVcsrsTQ0IUOXKeDs8Uhm8TBMWz1GTE4JaaGUIQA4ELBGvJeBy7/KZWeJ903+03bx+Ieh063p5QO3CgM3yQ+3oD56Wn+xZasCpv22sBvCwBcLkxL5nTmdvepCFLTTquFsdvFk2WoQyfYVkz7ZQkjt4opjdgzT/NT4lktJ3cmL4STZQDATcK19O/t6cxE1Rd5Uu/1QpcIdkZbtmsk9t/VmiCR7y9I/zwmVQrkL23Z0Q+xagxqHgCKEADcKlBN3mzHvtyKTS2Qxv8oBqnJy63Zp5pgb16NmB0krUD69wkpRk/+3oEb0JDB54gHhyIEABnoeDIlnn2xJbv+Ep17UnxtnzS5JTspjg3Xyp3MU50so1/kScvPSn2i2e96YV6gM6EIAUA2LEMGNWIGNeJPlNG5J6QWKx19o9lJcWyv+hjo3OKQyOqL0henpDMmMqkFe+IJVT0/uTP5HBQhAMivdTDz3+7cvx7mlvwkvbpXtDjIcy3YcbFslE7uZPI5epMuKpCWnpVaBTFT4tmhMSyPvceugSIEAE8RqCaT49nJ8ey+a/S/+VLLlY6HI5hnm7HDGrMGxVwh81oVWXZWWlQg3bSRcc2Z3BQel+ZxNRQhAHiczuFM53DusyRuzUVp6Vn6592O/g3ZZ5sxfaJ9dnrA9SrywwVp5XnpwDU6qBH7z4e55HoMZsS7B4oQADyUjicjm7Ejm5EbNm7lOemT49KY7WKfaHZoDDOgIRuskTufMxRa6PpL9PsL0oFrtH9DdnJLtl8fVocNs3vh9w0Ani5UQ15oyb7Qkr1hI+suSqvO0ym5jo5hzOMN2D7RTLtQLzuzRpBIbildf0laf4leraD9GrAvtmT7o//kg188AHiNUA0ZH8uOjyUVAre5SMouoiO3SjdttHc02zua6R7JNA/00E4UJHLoBv3xKs0plnZepU0DmAENmf924zqFY/+n/FxVhKIotmrVKj8/30XLBwAl8+PJ4Bh2cAwhhBRa6OYrdEsR/fshqVKgXSLYpEimawSTEMYEyHqKTZGV7i2huUXakweF3aU0Rs88Wo8Z35xN7YHpkp7FJUU4d+7cpUuXnj592hULBwD4rUZ65k+xzJ9iCSGkyEp3l9LdpXTmAenoDRqhY9qGMG1CSNsQJi6QaRrA+LtsL9j1KpJfTk8b6WkjPVlGD12nhJDEUBLvTybHs0uS2RCfOKjpk1zypmjbtm2zZs1SUlJcsXAAgD8S7c882YR5sgkhhEiUnDPTozfo8TK6/CzNL5fOW6hBRZoYmKYGpqE/idQxYVoSrmUidCRYQ4LVDMuQQPXdl2wViNFOjXZS/a+kkhZVkCtWWmglxRX0nIlKhMQGMi2DmBaBzP/FMYlhTAN/RhCEsrKq8HCDO38JUFsMpa66BjzD3L7wvLy8pKQk5m4Htl9++eUXXnjhtgdv3rxpMBhUKsVMICLEbrdXVFQEBQXJHcStysrK9Hq9ol5oh8NhNptDQkLkDuJWRqNRq9VqNHKOjEqq2MIKtrCCLarirtuYm3b2pp25ZmNNDqbcQSTKmAVGxRI/Tqr+/gqBcVCGEKLjaICKGngaoKIBPA3X0CitGKWl0ToxUksb6qQwjXTn0wmCYDQaQ0ND3bqScjMajRqNRqt16/7foKCgOr+1nDMijIuLq94Ret9ajYyM3Llz552P+/n56XS3X0OCYZjAwEBFbR9tNpvValXa9pFlWYPBoFb/wUdxX+RwONRqdVhYmNxB3IrneZ1O5+bt423CCWl9v++xS8TquDVd0V9FHuTGDoIg8DwfHh5e90V4IZVKpdFo7tykuxTL1v11ck4R1vykGJZla/6eYH9W11zeR4GrTBS51gpcZeI9a61lidZJR428ZZWdy+vW2muCAgAAuAKKEAAAFM2FRfjgp+HMnTu3sLDQKWG8xenTp7/++mu5U7jbV199VVBQIHcKt7p48eJnn30mdwp3S01NPX78uNwp3Orq1av//Oc/5U7hbkuWLDlw4IDcKWrBo0eEGRkZpaWlcqdwq6KioqysLLlTuFtWVtaVK1fkTuFWJSUlmZmZcqdwty1btly4cEHuFG5VVla2Zs0auVO4W05OzpkzZ+ROUQseXYQAAACuhiIEAABFQxECAICiufDKMnc6e/bsoEGDOnXqVMPv37p1a/v27RU1u7ykpOT06dM9evSQO4hb/fjjj3FxcREREXIHcZ8bN24cO3YsOTlZ7iBulZubGxMTEx0dLXcQ9zGZTHv37u3Tp4/cQdxq3759kZGRMTEx7nzSadOmdejQoW4/69YiJIT8+OOPSjtaDgAArpacnNywYcO6/ay7ixAAAMCj4BghAAAoGooQAAAUDUUIAACKhiIEAABF89wiLCsrS0lJCQkJGTx4cFlZmdxx3EcUxbi4OLlTuM+aNWtat24dFBTUo0cPhVxxNCsrKz4+PigoKD4+Pjs7W+447nPixAl/f3+5U7jPI488wvzszhuP+ypBECZPnhweHv7II48UFRXJHadGPLcI58yZExMTU1xc3KhRo48++kjuOG4yd+7crl27Vt/lWAkKCwtHjx793//+t7i4ePDgwRMmTJA7kctJkjRq1KjPP//85s2b7777rhJWuZrRaBw/fnxFRYXcQdyEUpqfn3/58mWz2Ww2mz/99FO5E7nJp59+ajKZLl682LVr17/+9a9yx6kZ6qliY2Pz8vIopXl5ebGxsXLHcZOtW7euW7fOk18X59q2bdtzzz1X/f/S0tLQ0FB587iBzWbLzMyUJMlkMq1duzY+Pl7uRO4gSdLQoUNXrlypnPd2cXGxXq/v0KGDXq8fMmRISUmJ3IncJCEh4ciRI5RSk8l04MABuePUiOfOI9Tr9deuXdPpdJWVlZGRkSaTSe5E7sMwnvu6uIgoilOnTmVZdt68eXJncQeLxWIwGBiG2blzZ9euXeWO43Iffvjh9evX//WvfynnvX3kyJFXXnnlk08+adSo0fTp0+12+7Jly+QO5Q6hoaGTJk2aP39+06ZN09LS2rRpI3ei+/PcXaOUUoZhqv8jiqLcccCFNm/e3KlTp8DAwLlz58qdxU30er3FYpk9e/a0adPkzuJy27Zty8rK+uCDD+QO4lbt27f/5SKRH3744caNG+VO5CYmk4lSevLkyX79+k2aNEnuODXiuUVYv379S5cuEUKKiooUdXFCRaGUvvnmm+++++7y5cs//PBDnuflTuRyFy5cmDFjBiHE399/4sSJeXl5cidyuS1btuTk5KjV6uqPttXjYLlDudyhQ4dyc3Or/69WqzUajbx53CY8PPzll1+uV6/e1KlTT5w4IXecGvHcIkxJSUlNTaWUpqamDhkyRO444BK5ubnp6elr166tX7++xWKxWCxyJ3K5+vXrL1y4MCcnh1K6YsWKhIQEuRO53OzZs385GEMIoZR269ZN7lAuZ7Vahw0blpeXZ7fb33vvvaFDh8qdyE0ef/zxRYsW2Wy2+fPnd+zYUe44NeP2o5I1VVZWNmDAgOjo6JSUlPLycrnjuJUnvy7ONXv2bG95QzrR9u3bExMTg4ODk5KSqs8IUw6FvMSUUkmS5s2b16xZs7CwsLFjxxqNRrkTuUlxcXHv3r0DAwN79Ohx5swZuePUiFIOXAMAANyV5+4aBQAAcAMUIQAAKBqKEAAAFA1FCAAAioYiBAAARUMRAgCAoqEIAQBA0VCEAACgaChCAABQtP8HpkF2eHuNrF4AAAAASUVORK5CYII="},"metadata":{"image/png":{"height":480,"width":600}},"execution_count":1}],"cell_type":"code","source":["c = pi/4; m = cos(c)\ntl(x) = f(c) + m * (x-c)\ntls = tl.(xs)\nplot(xs, ys) # redo plot\nplot!(xs, tls) # add tangent line"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

To make a parametric plot of $(f(x), g(x))$ is similar, though both the xs and ys are generated in terms of $t$ values. For example, here we plot $(\\sin(t), \\sin(2t))$ over $[0, 2pi]$:

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{"outputs":[{"output_type":"execute_result","data":{"text/plain":"Plot(...)","image/png":"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"},"metadata":{"image/png":{"height":480,"width":600}},"execution_count":1}],"cell_type":"code","source":["ts = range(0, stop=2pi, length=100)\nxs = sin.(ts)\nys = sin.(2ts)\nplot(xs, ys)"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

(There is also the interface plot(f1, f2, a, b), but we will not use this.)

","metadata":{}}, {"cell_type":"markdown","source":"

For a 3-dimensional parametric plot, the plot function is also used, the steps are identical. The helix, a plot of $(\\sin(t), \\cos(t), t)$ may be visualized through:

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":"Plot(...)","image/png":"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"},"metadata":{"image/png":{"height":480,"width":600}},"execution_count":1}],"cell_type":"code","source":["ts = range(0, stop=2pi, length=100)\nxs = sin.(ts)\nys = cos.(ts)\nzs = ts\nplot(xs, ys, zs)"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

How to plot the vector-valued function helix(t)?

","metadata":{}}, {"cell_type":"markdown","source":"

The output of helix.(ts) is a vector of vectors:

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["4-element Array{Array{Float64,1},1}:\n [0.841471, 0.540302, 1.0] \n [0.909297, -0.416147, 2.0] \n [0.14112, -0.989992, 3.0] \n [-0.756802, -0.653644, 4.0]"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["ts = 1:4\nhelix.(ts)"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

This is in the wrong order, as we need xs to be the first entry of each answer, ys the second, and zs the third. Some data reshaping is necessary. The following function xs_ys extracts values for xs, ys, and optionally zs, from a vector of vectors:

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["xs_ys (generic function with 1 method)"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["xs_ys(vs) = (A=hcat(vs...); Tuple([A[i,:] for i in eachindex(vs[1])]))"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

For our example, you can see how the values are collected:

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["([0.841471, 0.909297, 0.14112, -0.756802], [0.540302, -0.416147, -0.989992, -0.653644], [1.0, 2.0, 3.0, 4.0])"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["xs_ys(helix.(ts))"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

We add a few different calling methods, that may prove convenient:

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["xs_ys (generic function with 4 methods)"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["xs_ys(v,vs...) = xs_ys([v, vs...])\nxs_ys(r::Function, a, b, n=100) = (ts = range(a, stop=b, length=n); xs_ys(r.(ts)))"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

The first allows the vectors to be specified as arguments to the function, the second a convenient wrapper to the process of creating some ts over [a,b] and then corresponding xs, ys, and (optionally) zs.

","metadata":{}}, {"cell_type":"markdown","source":"

(This function is in the MTH229 convenient package.)

","metadata":{}}, {"cell_type":"markdown","source":"

With such a funtion, the above 3D graph could also have been generated through:

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":"Plot(...)","image/png":"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"},"metadata":{"image/png":{"height":480,"width":600}},"execution_count":1}],"cell_type":"code","source":["ts = range(0, stop=2pi, length=100)\nxs, ys, zs = xs_ys(helix.(ts))\nplot(xs, ys, zs)"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

The last two steps can be combined as:

","metadata":{}}, {"outputs":[],"cell_type":"code","source":["ts = range(0, stop=2pi, length=100)\nplot(xs_ys(helix.(ts))...)"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

Using \"splatting\" to create 3 arguments for plot from the container with 3 components returned by xs_ys.

","metadata":{}}, {"cell_type":"markdown","source":"

Finally, these two steps could be just one, through the alternate iterface:

","metadata":{}}, {"outputs":[],"cell_type":"code","source":["plot(xs_ys(helix, 0, 2pi)...)"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

As another example, this will plot the pringle function:

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":"Plot(...)","image/png":"iVBORw0KGgoAAAANSUhEUgAAAlgAAAGQCAIAAAD9V4nPAAAABmJLR0QA/wD/AP+gvaeTAAAgAElEQVR4nOydeXwURdrHn+6eSQIJd4Ak3DfIDQKCKJcCHngg6+u5rrKKN3i7q7h4cHsrKOui64UHHouIIIICIpfIFQ5BLg1ngHAFAkx31ftHkbLSPdPpmeme7pl+vn/kM+lMaqq7a+rXz/NUPY9EKQUEQRAE8Suy2x1AEARBEDdBIUQQBEF8DQohgiAI4mtQCBEEQRBfg0KIIAiC+BoUQgRBEMTXoBAiCIIgvgaFEEEQBPE1KIQIgiCIr0EhRBAEQXxNwO0OIIgXIYSw7IOSJLGf7AWCIKmHhLlGEYRDKaWUapomyzJTPvYF4ccVRYFw6mh8gSBIsoAWIYKcRVVVQoiiKIFAIJKeBYNB8dmRyyQhhP8KgkaK7RiPIAjiBVAIEQQIIUzJZFlmNp8JRm3TwdURBGkEwd0KZUVRVE1KKSolgiQYFELEvzBLjhDC9U/TtPibLdfyE92t7LXOoASDUlppFkGQ2EAhRHyKpmnM/AoGg+xIwuLlov0H5ZmVol7q1u8YG4zUGoIgJqAQIv5CZwV6VjZMVM3odOVmJbMsRV+rrh3Pni+CuAgKIeIXRAnkVmAyYqKRmqZx8RP1kgkklF3OA+GClAnoP4J4DRRCJPVhOx8AINkl0CJWVE10txpXvQIGKRE/gUKIpDJMAimliqLIMuZR+pNyUwSI7lZZljFIiaQwKIRIasJmcJTAmGF6Jssyzy3AMQlSGl2vGKREvA8KIZJqiJsC+U/ERqzvDwFDkJKJJRfXsKFKBEkwKIRI6sBjgXw5KJ9/kURirpShUIgl6DHupzTuJ8EgJZIAUAiRVEC0AtEETApM7L+wSex0/lhMYofYCAohktxwCfTypkAkKsyT2HEtFBO9Ml0U175GcrriIEGMoBAiyQpKoD8xSpokSTo3gC5IKe4P0SX0wSAlAiiESDJCCGE7x1ECkbDEsJPSuBQWd1L6BxRCJJnQNI1tazOplIQgVjCx/1RV5VYmBin9AAohkhwwCaSUBgIBXA6DOIpUCmClLX+AQoh4GjFBaCAQYMsi3O4U4nei3UnJBJKHJ42NoOvVXVAIEe/CHaF+SBCKpBI6SQtbbwsrbXkHFELEc/AnaIwFIibMmzfvkUcecbsXTvH0008PGjRItCnZcZMgJa56jRkUQsRD6ByhbncH8TSHDx/Ozs6eMGGC2x2xn3/9619FRUXmOQdUVRWX8wAAIQS/NbGBVw3xBLxMBDpCEetUrVq1Y8eObvfCfqpXr17ue4y7J5GYQSFEXAYrJSEI4i4ohIhrkFJwRwSCIC6CQoi4AC8WyGKBqIIIgrgICiGSUHQJQvmqcQRBELdAIUQSBFZKQhDEm6AQIs4i5v5HCUQQxIOgECIOwqxALBOBIIiXQSFEHAGLBSIIkiygECJ2wrOjAUoggiBJAgZsENvQNE1VVVVVFUXBHKEIwtE0rWXLlm73AokICiESLyw1jKqqAMA2BaIEIgjnlVde6dGjx+bNm93uCBIRFEIkLrgEKoqCvlDE59x+++0vvfQSez106NAXX3wRANq1azdy5EhX+4WUAwohEiNMAiVJCgQCKIEIAgDXXHPNl19+CQCnT5+eMWPGddddBwB9+vS5/PLL3e4aYgYKIRI1mqaFQiEAYGmyUQIRhNG3b9+NGzfu379/7ty5HTt2zMvLc7tHiCVw1ShiFayUhCDmpKWlXXbZZV999dWCBQtuuukmt7uDWAUtQqR8WBVQTdOYBCqK4naPEMSjDBky5MMPP/zuu++uvvpqt/uCWAWFEDGDrwjFMhEIYoWLL774l19+6du3b+XKld3uC2IVdI0i4eGpYdhyGAwEIogVMjIyzjnnHKNfFKuseBkUQkSPrkwEfoERxCKhUCg/P7+goKB///5u9wWJAvR0IX9CCFFVlRCCjlAEiYGZM2decsklkyZNSktLc7svSBSgRYhgpSQEsYfBgwcPHjzY7V4gUYNC6HewUhKCID4HhdC/YKUkc1hwFK8MgqQ8KIR+RNM0tAJNYL5iAJAkiT0rMHiFKSgVSBRLBEkBUAj9hbgpEKdvHTxWKkkSD5SKV0l8zd4sSRJXTfYGdlD8F90RBEG8BgqhX+CxwEAgAGjElIWWAgAma4UkSTLqok4dxQ0nvE2dHSm+H5USQVwHhTDFYSYO2xHBHKGifw8RvaBMk+JpTSeNRnkzts+eTsTj4r+jRiJIAkAhTGVYjmxJkjBHthGdBEJCcn8YVU38dJ0iMo0Ug5RhpdG3SpmWlpafn3///fe73RH7WbFiRe/evd3uhY9AIUxBjFag2z3yFkYJ9AJGR2tY16vxV1EpeeRSbCRVzcorr7yyuLj40KFDtp8gewSxsc1oF1U1bdq0W7dudn06Ui7x+oIQTyFWSopUI4K5Rq1XkIj2OxwKhawboGzitr6Fn1XAsN4Z9jSg+7hIcxy7eiyGGpaoTs0Kti86ZSuheIPGb3dYk9RESJxYFmv7ZdTd5fjRNE1cMBU/tl/GcscqEhV4HVME9sUATA0TAVECU9I8YpgEKY2uV0ZYByx/karWJIKIoBAmPdwKZPXi3e6O5xBXbPp5Tjc6WnXHwwYpoVQp+Zv943pF/AMKYRLDU8MAAC6HMWLuCEV0hA1S6i6glf0hqJRI0oFCmJToKiWJfi0ESudlVEHbsbI/ROePDauUYbeLIIhboBAmGRgLNEfcF68LfSEJwGRrh0mQMlIjaE0iiQGFMDlg8ztWSjLBJ8thkhduTepukOh3jWEnZVhxRZCoQCFMArBMhDl8HsTng6TG4k5KY5CSPyNChNVAaFYi5qAQehqUQHPE/eN4ffxApCAlfwaK5HrFJHaICSiEHgUrJZnDJRCtQEQk0v4QiMb1alzg42ynEbdBIfQcrFJSIBDASklhwVggEhsWXa/GX60EKXEoJjUohF5hz549JSUlDRo0YPO79RRoPoGHgjBvAOIcuhU9FoOUYGo+olJ6H5xQPMHrr7/euHHjNWvWBAIB9IXqYPrHHaF4cRAX0SmlLBDJUc9GLxHga3wgITVPkHJBIfQEAwYM6N69Oxo6Ovh8gRKIeB+jNakTS+Mw5knsRI0UlVKXfABxCHSNeoJmzZpVq1bN7V54CP79x+UwSMpgY5ASyn5H0PUaJyiEiLfAtTCIn9GNeZ1S8q8Gl0b+feHZNhLb3xQBhRDxELwgKkogghgRd0Dqvin4lYkHFELEfShWSkIQxD3QjkbcRJccC1UQQZDEgxYh4g484I/LQREEcRe0CJFEo9sX6HZ3EATxO2gRIokDV4Qi3oQCzC6gX/1OMgKQocBV9aFrTbf7hCQQfB5HEoFYTBFVEPEOGoWPtpEOX6hPrNRaVZUaZkkZClw5j0zehNvYfQRahIizoBWIeJajZ+C679UjZ2BcF2VgvT9H5/WN6HUL6OEzZGRHNBV8Ad5mxCnQCkS8zLZjtMdXav0sadHlgUvqlRmdTStLX/aTXl6vFZa41j0kkaAQIvajk0C3u4Mger7fQ8+fqd7fRp7SUwmGmwXrZ0k3NJHHr9MS3jXEBVAIEdvglZIAJRDxMAv20ht+UD/uGxjW0mwC/GcH5b9byMFTCesX4hoohIgN8B0RgBKIeJt1RfSGH9RP+wV655YzSnMrQs8c+Ye9xPxtSAqAQojEBRYLRJKIbcfopd9qr3VXLsyxNFAvyJF+3IfLR1MfFEIkRlACkeRiz0naf7b2dCf5mkZW570Lc6RFe1EIUx8UQiRqeCAQJRBJFk6ocOkcbVgreWiLKCa9TjWkncX08Gnn+oV4AhRCJDpwRwSSjAxbrHWoIT3aLroZLyDDudnSzwfRKExxcEM9YgmslIQkL69vJPlFdOkVsUx3DStJBcUUAMd8KoNCiJQD2xSBLlAkSVlWSJ9brf00KFAxptkuryLsPml3nxCPga5RJCK0FFRBJEkpLIG/zNfe7Kk0qRzjAM6rKO09ia7RFAeFEAkDVkpCUgCNwk0L1FubS1c1iH0M51WE3Sds7BTiRdA1ipSB58hG/UOSndFrCAUY1UmJp5E6mdKek7inPsVBIUTOgmUikFRixQE6eaO26uqAHN9YzqkAezFGmOqgECIogUiqcVKFvy7QXu2u5FWMdzxXCEinNIwRpjgohL4GHaFISvLgMq17benaxjaM6oAMKnpGUx0UQp+CEmgCIYTnEOdWsiRJ7Dh7zX8iXmPOLjp3N10z2J7JLSCBigZhqoNC6C94pSRZllECjRBCVFUFAHZ9mBbynwwoTa/DlVL8yV+w5wzXzsSvHDgFQxdp0/oolYP2NBiUIYQWYaqDQugX+CQOWCkpHJRSVVUJIYFAgD8i6Cw/QoiiKLr/grJKyTWSaSEYlNLYLGIjf/9Ru6WZ1Ku8EkvWQdeoH0AhTH24BDJ7hTv9EAhXTDgqQ9mKpOkMSlE7GeInolLGw3u/kT+K6fR+dk5rEoAig0oggA6U1AWFMJXhcy4uBw0Li/kx8WOOUE3TbP8UE1VjAizCXa9iclco63QVjyOcfSXwyArtmwGBNLsVS5FAozhXpjJ4c1MT0QrEWKARvuxFURQXFaVcy0/negVBO41Kyf7KFN2HMnnvEu3vLeTO2b47cSR+UAhTDTEQ6G5PvImmacwKdFcCLWLR9Sp6XzVNE+PB4I8g5Wc7yIbD9IPe9k9oKgGNQnpc2WkQr4NCmDrwHRGQWnOcXTAJlGU5EAik0vXR+b2NPgCTIKVxtCSjUh46Dfcv1ab3C2Q4IFcnVMjEaTLVwTucCuiWw7jdHc/BY4EpJoEWMVE148odXZBSTDnk2SDlA8vIdY3l82s70p8TKs0MeOVMEYdAIUxudIGiROL9dXTMT6hpmizLSeEITTzlWn7ich4rQUrd0p4EMLuALjsAawc75bs8EYLYChkiSQTe4WSFP7AnflPghsP04tnqvpNwczP5le5K1bREfrhVmCMUAIJBm3ZW+5IYgpRG16uuKTFrT5zdOxaCO3/S3r5Ads57ia5RP+DtR3okHNTtYoEPLdcebqscuyWYFYT2X6hbjnpoVyKzAkOhEAAEAgHd/nfECfj+E0VRFEUJBAKBQCAYDAaDQUWAmYls9GqapqpqKBQKhUKqqjLDnQjo1vtE4p8/awPqSn1yHTy7Eypk4qNUqoOPOskEXw7jYiyw6DQsK6Qz+8tBGSb1UM7NJpd+qy29IlAzw5Xu/AmbYdlyGG4FYuoAdzFahADAlJK9Nu4PYVhJYrdsP/liJ9lwTRDAwdQvaBH6AbzDyYF3NkVsP06bVJaCpb24tbn8ezEdNFf94dJABZdGE98IL0ogkhREu5OSpwEKaXTYYhh/LmTJIVX9M0Bge5DyRAgXy6Q+KIRex2ubInYcp40qlenGvzop249pd/2k/bdXov2QfNucoiiuPyIgThBp5L+8ieRUJDc3DwAAWw9lsj/E2JT1L9ThM1A13a6zQTwKzh2uMWvWLPM36HJgekEFAeDoGahWdnWMBPBmT2VpIf1iZ+KSE7MyEaqqMisQVdBX7DxOx6/VJp9/9sGLfTtsD1KyxveXQO0K7p0qkhBw+nCHdevWvfXWW5H+yiXQO/rHSVfgtCEfZ8UATOuj3LtE21fieAeYBDIjACXQn9y/lDzQRmlSufyvhrgJkq/oEZWSrajiGslhS65CodDeE1rNdCqKpcWFPEgSga5RdxgxYkRmZqbuINc/8EAsMBIZCpwKl5i6c7Y0tIX8t4XqV/2c+mhKKTMB/bkvHmFM30G2H6efXWSPHz6Sj5QvyT5wWutWC2RZAkOlrbCNJGNqHsSjs21q85///CcvL088smDBgnXr1kHp18/L359IQggAT3VUDpyCD7bZ/6GiFYgq6GeOnoEHlpEpPRXbS0xEorCE1q4gR9ofIjpg2bDUuV6ZA18TiGp/CJIY0CJ0gRtvvPHSSy8dNmwYABBC+vfvX1BQ4KlAoAmVgtKxCBW7gzL8u6dy+bfq1Y2gmh3rC7iTivu1nCiThCQRT6zULqsnOZRNLSyFJVCrvBhhuUnseKRDPGJMzePZJHYpD1qELlChQgXu+ZRlefz48Zs2bWrTpo27vbJIvUwoKI74187Z0lUN4MlfbJAr9ljNSsaLVeMR3/LzAfrFTjKuS0IXJ+8vobUrxC5IYnhSZ1aKQUr2HMyf/BihUkSzUreWB7EFtAjdp3Pnzm53IQrqZkq7T1JCQY4wOTzdAdrPILc0k7vWjHH6YBIoJUmlJCQxEAr3LdXGd1VscTZYRCVw5AxkO5kswjyayAVPfMFQVVW0KbniOtjX1AWvGhId6QpUS4N9JREfSKulw5guyj0/aST6Z1b2FEwpZVYgqiDCeWMTSZPhpqYJnbIOnILq6aC4Nwx1S16Nq155kBK/LPGAQohETf0s6Y/I3lEA+FtzOV2BqZuj2FaoSxCK32pEZH8JPLNae7NnoiWp8BStFYdfNGFwpXS7I8kKXjgkaupnSb8Xm5l7EsDrPZSnftGOhcppikVEVFUFlEAkMg8v125tLp9TNdFjY/cJqFMxwZ+JuAAKoTvUrFnTZEO9x2lZFTYdKcfv2aGGdEk9efxas1UzKIGIFX7cRxfto092cKGQiDGhIJKSoBC6g6IoOTk5bvciRtpWk/KLyn/bc+fKUzaRPwy2I7cCJUnCSkmIOWcI3LlYe7W7nOVGNvXfi2mDLBTC1AeFEImattWl/MPlr4TJqyjdfY488pcyJc65FciXjDvYUST5eTGf1M+CKxu4M1PtPA4NK7nyyUhCQSFEoqZZZWn3CXpCLf+dj7ZT5u2mKw9SniMbAHCRG2KRP4rpC/napPNd8xnsLKYN0SL0ASiESNQEZGhRRdpowSjMCsJTHaVHlv2ZIxsdoYh17l9KRrRRGrsXpdt5nDbEGKEPQCFEYsGKd5Tt+b2lCTlwWvp2L6aGQaLj6z/opiP04bauDZsTKpxQy8+vhqQAODchsdChhrTqYEQhFHNkZ6QFx3aRn1gZy/56xLeUqDB8qfZ6DyXdPQ/CzuO0QRZ68H0BCiESC91qSssKwygbk0BCCMt/wazAQfXlrCB8vD1xZXuRZGfsWu3cmtLFddyUoZ3FuFLGL6AQIrHQOVvafJSeFNbLcAkMWylpXBflyZXkDEohYoGtx+ibm8iL3VyenXYex5UyfgGFEImFdAVaV5N+OXi2TqkogWFjgRfkSC2qRJd0DfEt9y3RHm2v1Ml0WYRwE6F/QCFEYuS8WtLS/UTTNEqp6AiNxOhzldFryEkLmy4QP/PFTlJwAoa3dn9q2noMmlZ2uxNIQnB/tCHJCCGkS3WyrJCKtbnN6ZQtda8lvb4RjUIkIidVeGg5ea2HEvTAzLTpCG1VDS1CX+CB4YYkDyw1TCgUIoT0yFGWHaBR7Ysffa78/Drt8GnnOogkN8+s1nrWlvrkui8/IQK/F9MmuInQH6AQIlYRc2QHAoFGlWUAMC9DoaN5FemKBvKL622oX4+kHpuO0Lc3k4ndPJFyYdsxWi9TcnHzBpJIUAiR8mHLYcBQJqJXrvzDnui2Bz7VUX5jIzl4yv5OIsnOvUu0kR2VHG9sYN98DFomvOoT4hYohIgZvGR82Fhg31zp+yiFsH6W9H9N5Inr0ChEyvDxNnL4NNx9jldmpE1HoGUVtzuBJAqvDDvEa+gqJYWNBfbJk+ZHKYQA8GQH5e0tZH+JHb1EUoLjIXhkBZl0vpLoCvSR+fUIbYUWoW9AIUT0MCsQLFRKalpZCsqw5Wh0WphbEW5AoxAReOoX7ZJ6UvdaHhKeX4+ia9RHoBAiZ2E5spkEWq+U1Ccvau8oAPyjg/LOFrL7BKYfRWD9YfrRNjLmXG+tS9l8hLaogkLoF1AIkbMSGFulpD650g97o9aznApwa3N5wjrcU+h3KMC9S7TR5yrZGW53RWD3CZoZhGrpbvcDSRQohL6Gl4w3yY5mzkV1pO/3kBgqSzzeXpm2jexCo9DfvLOFnFTh1ubemoh+PSq1wJwyfsJb4w9JGLxSkiRJwWAw5mKBeRWl6unl1yY0kp0BQ1vIY9d6VAgpPZtGlb1wuzupSdFpeOJn7d89FdljPsjNRykGCH0FCqHvsJIjOyoG1JVmF8QiFY+0Uz7dTqLakp8AmP6x1/zikHBQA+71Oin558/atY3lDjU8Jzn5h6E1JlfzEyiEPoLFAm2UQMal9eRvCmKJ9tVIhztbecgo5CYgWyvL1gqxF7IA+9W4koj9e1ilZH914ZQ8zIoD9OsC+kxnb62RYaw5RNtXd7sTSAJBIUx9eKUkthzGRglk9MmV1hXRQzFlEH2orfy/3+mO4y6LBLcCmciZv1kUSBETpWSN6zTSaFM6eYreglC4b4k2oatcJc3trhjQKGw6Cu2qo0XoI1AIUxxCzlZKckICGekK9MqV5+6KxSismgZ3tJTGrXVt+ShXINEKjBOjUnJ15PC38W5AeUoJKWRWvrGJZAbghiZenH+2HKU5FaRKQbf7gSQQLw5ExBZ4LJBlR3NCAjmX1ZNmxRQmBIARreUvdroQKRStQFv0LyqM1qRRKcVbxnXRPEKZFEq59yQ8vUqbdL4XnaKAflFfgkKYgvBKSaxebgJm+cvrS3MKiBbTJFw9He5sJY9ZkzijkEuged4cd9GZlcYgpfFfwjpgPaiRj6yEO1rKnk1ghkLoQ1AIUwqeHY1VSkrYLJ9XUaqbKS0vjHGqfait8uVOkoBIobtWoF2YBCl1P/m/WAlSQkLEcu5uuvwA/KODR81BAFhziHpwISviKCiEKQKvlGQ9O5q9XFZfmhXT2lEAqJoGd7aSnYsUco8i2BoL9CwWg5SJd72e1uD+Jdor3SAzEN8ZOsnaItq+mlesZyQxoBAmN7S0ZDwtrZTkVk8uqyd//Ufs08cDbZQvdpKddhuFfH6PtOfBt5i4XrlS6rCilOaMXqO1qy5dWtf+07GLvSdBo1AnE8eJv0AhTGI0TdM0DdyzAkXOqyUdPAWbo6xEwamWDne1ksfYZxSiBMZJuftDjP9iknMAALYcIW9uIi+d5+k5Z/Uh2hH9ov7D04MSCQu3AgHAXStQRJbgqobSlzvjMgptiRSiBDqNSZAy0k5KjZA7fyJPdpBzK1Av7w9ZW0Tb4w5C/4FCmEzQ0hzZ4A0rUMeQRvJnO2I36ZhRODqO5aO07L54T10cX2FUymnb4VgI7mmteHx/yNpDtD1ahP4DhTA5oKWVkiRJirZSUsK4MEcqOEF3Fsfewog2yv9iMgp1y0FRAj1F0Wl4bIX2Zs8yBejjD1I6sT8EXaP+BIXQ6zAJjKdSUsJQJLi6gfz5zthbqJ4Od50jj40mUsjnQZRAz/L4z9qQRvK52dHdmqhcr7YksTtyBvadxLoTfsS7sypChASh8VRKSiTXNJK//COuFh5oo3yxw9LyUbQCk4Il++k3BfQ5WwvQW9wfIkWZxG55Ie2cLSk4jvxHEsytPoTYXSkpYfTJlbYfhz/iyJd2NtGMqVHIZy4MBHqcMwSGLdZeOk+unPDUnUZr0qiUuiDlskLStSaISpn4ICXiCkkzw/oEWlopiWVHSyIJZARkGFQPPo9j7SgAPNhW+WJH+EihbkVoPJ+CJIAJa0m9LPhLIy/eKWOQcvkB2r12OftDnA5SIq7gxQHqQ86cOXPo0CG2L5BZgclr6AxuAPGsHQWA6ukwzJBohkZTKQnxApuP0lc3aG96Nbm2Dgrw8wHateafPtXEBykRt8AJxROMHDly5cqVSecIDUvfXPjtKI0zR8yDbZXPd5wtScFnENkH2dFSBgowbLH2ZEelflZy3K9tx2hmQMqtaOnNDgUpAc1Kl0juOTdlGDNmzIABA1Jjig/KcG1j+cNtcX2fa6TDHS3lcWsJe9CGsnMK4n2mbiYnVbjnnKSZYZYV0vNq2TnGYghSemEnpT9JmmGa2nhzX2DM3NxUfvc3Es/3lVL6QBvp0+3kjxO4IjT52FcC//xZm1J246DHWV5Iu9kqhOViDFJG2knJxU9nR2KQ0i5QCBH76VZLkgFWxFSViccCszOkO1rKE9a5VrweiZnhS7XbW8rJtTN9+QHataa3OqyzJkWxZG8QxZL7TpAYQCFEHOHmZvL7W6PbF8/cQSDEAh9qq0zfQQpO4HNuMvH1H3T1ITqyYzI5OU5psPEw7ZQ8ys3dJDqxdLtfyQoKIeIINzWVPtlOzliQQho5R3Z2BtzewsE6hYjtnFDh/qXaG+crGcmkg7DqIG1VVarg4SqJiKOgECKO0CBLal1Nmm1aqpeHN0yeZx9qp3yy7ezyUcT7/ONnrU+u1C8vyUyT5QcSHSBEPAUKIeIUf20mv/dbeAHjViCUOkIjNcKWj45HozAZWF5IP9tBJnZLKmMQAABWeC9AiCQSFELEKYY0kr/fQ4pOlzlIo6+U9FA7jBQmAac1+PuP2kvnKdXT3e5K9Py4j/bMQSH0LyiEiFNUDsLAevIn2/805mLLkV0jHW5vIY+No04hkgCeXa01rSz9X+Pkm1K2HaMyQONKKIT+JflGLZJE3NpcnrKJiLHA2DYFPtRO+XS7pZIUiCusOUSnbiZv9Ew+pygALNxHe+WiCvoaFELEQS7Kg5MaLC+MXQIZNdJhWCvcU+hRVAK3LdKe76bkVHC7KzGxcC+9EP2i/gaFEHGEs9kuKP17C/nfm22w5B5si0ahR3l2tVYnE25smqyTySK0CH1Pso5dxLPo9gXe1lz+3+/6JTMxwIzC59fb0UXEPtYW0Tc2kTeSpMSEkYIT9JRKm1dBIfQ1KISIbfAVoWIWqOwMuKSu/P5vNng1H2yrTN8Rb10LxEZUAkMXaRO7KXUzk1VIFuylvXJxGvQ7OAIQGxA3RRgDgXe2kqf8GlcObkaNdLijpTRuHQqhVxi7ltSuALc0S+JpZBEGCBEUQiROdBIYdjnMBTmSIsGivTYI2ENtpC93ku1oFHqAdUX01Q3am8m5UpSDS0YRQCFEYkZ0hJa7IvT2lvKUX23wjulTUtcAACAASURBVFZNg7tayWNwT6HbMKfo+K5KvaR1igLA3pNQdIq2qprEp4DYAgohEjU6CbTyL7c0k+fsIoUlNnz6A23kr34nm4+iUegm49eRKmlwa/PknkAW7iUX5soy6qDvSe5xjCQSXgsUopFARpU0uLqBPHWLDZZclTS4v7UyejUaha7BnKJvX5hEZXfDs2gfBggRABRCxArWc2SbMKKNPGmjpcJMVpr6bjf59QgahS5wWoObF2jPd1PqZyW9hCzcS3tjgBBBIUTMMSkWGC1tq0utq8K0aKr1RiIrCMPbKM+gUegGT6zUmleRbk7a7fOcfSWwr4S2rY5CiKAQIhGwUiwwWh5qp7yQb8M+CgC4r7X8wx6yrgiNwoTy4z76yXaa7CtFGd/tJv3y5KR37yJ2gEKI6LHFERqW/nWkgAzf7rJBvTID8FA75Tk0ChPI0TNw8wJtSk+lRhIWWjIydxftXxdlEAFAIUR0iKlhbJRAzgNt5BfyNVuauvcceWkhXYtGYaK4Z4l2RQPp0nqpIB4UYP4e0r9OKpwLEj8ohAhAWUeoE/rHub6JvPkIrD5kg3plKPBQW/npVWgUJoIvdpJfDtJxXVLBKQoAaw7RKmlSCqz3QWwBhdDv8B0RECFBmr0EZbjnHPmlfHvU685W8s8H6M8H0Ch0lj0n6d0/ae/2UioG3O6KTczdRQegXxQpBYXQv/BYoJXUMDYyrJU8q4DsOmGPUfiP9vLTq+zxtYpQAdsbTy4owK0LtXvPUbrWTB3lmLubXFwHZz/kLDgU/Aif4sUyEQmjahrc0kx+dYM9RuHtLeVNR2DRPtvkilKqaZqqqqqqapqmaVooFAqFQvxXTdNIKX6QyVfXk+Mh+EeH1JkrTqjw8wHaC7fSI6WkzuBGrCDmyE68BHKGt5Hf2UIOx12kEACCMjzVSf7XLzYYhVwCASAQCASDwUAgwF6w13wNEU+yI8qkqJRcLJPdptx0hI5ZS97vraTSNoOFe2mXmlJW0O1+IJ4BhdAvmFdKSjANsqSrG9q2fPSmpvL+Evhud+x6I0pgMBhUFCXsJWKXTpZlRVEURRFlkh2RBJgEsmZ1YilqpJdlskSF638gY86Vm1ROIRkE+HYX6Y9+UUQAR0Pqw2fbBMcCzXmyg/zGJnLglA1NKRI83Vn+589aDJJCKWXiJEkS07PY+sA1ksPFkhuUiqKIBiW3HUOliGalF1yvw5dpLatKQ1uk2iwxdzeulEHKkGpDHBHxlBWoo36WdG1j+eX19kQKhzSSCcCM36NojUmgqqqyLDO3py09iYSolKJByZWSyTDfvqlzvZoopUNi+el28sMe+u+eqTZF7DpBD5+m7Wt46+uAuEuqjXKEEXOZiETyzw7ylF/tqc0kATzbWXlyJbGSwI0QwuREluVgMOhirFRHWKXkBiWzKbnblislc73aG6Tceozet1T7tJ9SOeUCabML6MV1sPISUgavTAGILYiVkjxoBeqolynd0ER+wSaj8NJ6UrV0+GibWWtMAgkhibECbUSMPjodpDytwbXztVGdlI6paDbN3Y2Z1RA9STMRIOYYE4R6XAUZj7WT3tlC99thFALA6HOVUatIKJwUMkcoIYTpRxJJoEVsDFI+uExtUgmGtQDXg5S2c4bA/D1kYN1Uu/tInOCASHpsrJSUeOpkSjc2lSass2f56IU5UuNK8I5Q/pddHO4IDQQCyXV9bMRikPJ/f0jf7oYp50th94ckMkjpBD/soedUlWpmuN0PxGOgECYxSS2BnMfayf/dQnbbkWgGAMZ0UZ5ZTUpUAAA2j6ewFWg7BSfg3qXkg95K9QrhXa+iA1YXpBRzDhABTynljN/JlQ1wGCB6cEwkJakhgYycCvC35vL4dfZECjtnS91qSm9s0kRHaFJfn4RxhsCQ+do/Oyjn1Yq4h1JnVuqW84gaydFl5xHFkmtkYpSSAsz8g17RAAcDogeFMMkQl4MmuwRyHm+vfLyNbDlqw2yoadpT7cnEdeQkQQmMjsdWaHUzpftaxzgnGF2vuhU9YpASSpMYMI2MpJT2BilXHaRZQWhRBYcEogeFMGlIouWg0VIzAx5trzy8PC6jkE2mANAuO9C/rvy8TQUufML0HWTG73TqBQ5mUosUpGQaqVv7CoadlPEHKWf8Tq6on1JfHMQuUAiTAP5V574pt3tkP/e3ljcfpbEVr+cSyL1zo8+V39xE9tm0GDXlWVdE7/5Jm95PqeZ26fly94fEFqRkjf/vd3pVQ5zxkDDgsPA0fpBARpoM47vKDy7TVMuGHPOt6SSQ/alupvTXZvIzDpRnSj2KTsPgedpr3ZXO2Z4eXfEEKUOh0NbDocIS2rmaFjZIifgcFEJ3CIVCCxcuXL16daQ38K9xauufyFUN5LqZ8O9fLSlhuTmyn+igfLaDbDqC05wZGoUbf1D/0ki6rknSTwUmQcpgMPj1bmVQfTkYCB+kdCWJHeIdkn70JyM7d+7s27fvp59+etttt23dulX3V92KUFd66BYvdVeeXq0VRS7PZLQCI72zWjo80FYZtQqnMDMeXq5pFJ47N8ZU40kE2zgRKUgp7qRMTBI7xFP4a571ApTSa6+99uWXX540adLDDz/8+eefi38SV4S610fXOKeqdHUDefSaMC5Naq1SksiI1vLyA7C0ECep8HywlXxTQD/tF0ilWoNhKToNqw7Svnnln2diktghXsOPs627zJ8/v1OnTp07dwaA9u3br1+/fteuXcOHD58zZw54O0d2Ynj2XOX938hmYSsFLS0TAaUSaLGpCgF4soP0+AqMFIZh9SH60HLti4uUqmlud8V5vikg/erIFQNxNWJjEjvvVNpCGCiEiWbhwoWDBg1ir7Ozs48cOXL55Zenp6d37drV5xLIYFspHllOoGylpNiKBf6tmXTgFMwuwLmmDPtL4KrvtCk9ldbVfDHkvvqdJmDjhMUkdt6ptIVwUAgTzZYtW1q0aMFeHz16tFKlSqtWrZowYUL16tXd7Zh3uL+1vPUYnb7tTwmM2VEckGFMF/kfP2tWyjP5hDMEhsxXb20uXeWPZGOnNPhuN7msvidO1iRImeBKW4iIJwaHr9i3b19ubi57vWXLlubNm/szHBgJQohM1Ck96IjlcJzYUCzwqgZyZhA+NC3P5CvuX6Jlp0v/6pT6C2QY3xSQc2smQaJt3f6QsHl5uEHJYEIoKqXbJ5Gs4BScaNLT00tKzu70/umnnzp06OBuf7wDLxaoKMoFecFrG0sPLLMnvDe+i/LUL+Q0xgoBxq8lSwvpe71Tfn3Mn3y0jV7XOOknOuM2SlZNRbecx+1uJitJPz6SjpYtW27cuBEAioqK5syZc9lll7ndI/fR1ctlfqHnzlV+2k9tCe/1zJHaVpMmb/K7UfjBVvLGJvLNAKVSytWdj8TxEHy3mwz2R0IZXGQQM74YH56ib9++U6ZMmTFjxsCBA8eNGxcM+mZOMsBCIKFQKGzJ+MwA/LuncudP2rGQDZ81rqs8fq12OPIOxZRn3m768HLtm4FKnUwfTZczficX5siup45DPA4KYaK56qqr2rZtu2LFivfff3/gwIFud8c1eLFA5tsJGwvsmycNrCs9Zsf+h3OqStc0kp9d7VP3aH4RvXmB+lm/wDlVfaSCAPDRNnJ9E3+dMhID8e2sQWLi8ccfd7sLbsIi/ADAQhrmC94mdlPafq7O200vqhPvdDaqk9L6s9CdreTmPivEs+sEvXyu9noPpWeOv0788GlYsp9+0hcf95FywCGCJAiegyOqermVg/DG+codi7XiuB2kNTPgkXbKYyv8FSk8dBounq092Ea+ppHvvuzTd5ABdeUs/wYfEKv47ruBuALPjhZDyfhL60nn15b+8bMNXs37W8vriuiCvX7ZdFWiwhVz1SsbSMPb+PGb/sl28n+N/WUEI7Hhx68HkjDEBKG6SklR8VoPZfYu+tmOeI25dAXGdZUfWOaL/fUahZsWaA2ypLFd/Liqfl8JrD5EL6mHUxxSPjhKEKewRQIZVdPg077KPUu0rcfiVbAhjeSsILz3W4o7SCnAPT9px0P0v718tGVQ5NPt5Ir6coYfnwGQqEEhROyHSaAkSeaVkqKiU7b0dCfl/36gp+JzkUoAL3ZTRv5CTqRuFg4KcN8SbV0R/fyiQJpfv+IfbyMpUGQRSQw4UBDbEIsFismF7eLOVnLbatLwZfEac11qSr1zpQlrU3MrBQW4f4m2+hCdc0nAPxvndfxRTLceoxdZqLuEIIBCiNiCsVKSQ0kuXu8u/biPvr81Xi0c00WevInsOpFqoUIKcO8SsvoQnTMwUNmvKggAH22j1zSSAzi9IdbAkYLEBZNATdNirpQUFVlBmN5XfmiZtuFwXBpWL1O6q5X8z59TKlJIAe5dSvMPw+yB/rUFGe/9Rm5qipMbYhUcK0iMiMUCI6WGcYLW1aSXuyuD52nH49tZ+Gg75fu99OcDKWIUUoB7l2j5RXRWf9nnKrhkP9Uo9KiNflHEKiiESNSwHNncCkx8GakbmsgX5kh/XaBpcahYVhCe7Sw/sCyeNrwCBbhzsbauiH7jexUEgLe3kKEtZJRBxDoohEgU6MpEuFhJcfL5SolG7/4prgUvtzSTVQofxB1xdBdmC244TL8ZEMAsKsUh+GInubkZzmxIFOBwQSzhHQlkBGWY3i/wy0H69KrYZUyW4LXuyuMriC0FLlxBJTB0kba+iM7xfVyQ8cl20jtXzqngdj+QpAKFEDGDVUrylARyKgVh1oDAB1vJlF9j18IuNaVL60mjfknKrRTHQnDZt+qBU/SbgWgLnoX5Rd3uBZJk4IhBIsIqJVFKWYJQ70ggp3YF+PYS5dnV5PM4sq+N76pM20byi5IsVrj7BO31tdqwkvTlRYFMrCIDAACbj9I/imFgXYwPItHhuakN8QLcClQUxblNgbbQuJI0e4By7xLtx30xKln1dBjZUblnSTKtmllbRM+fqV3XWJ7SU8Hdcpy3fiW3NJN8mlMOiQP8DiFlYKlhoqqU5Dptq0vT+gaGzFfXxmrV3dVKPqXBp9uTY9XM/34n/Werr3aXH2uP398/OUPgg63kb83xmiBRg4MGOQvPjsZKxieFBHL65EqvnKdcOkdbfSgWLZQlePk85dEVNP6qh07z0npy3xIye0Dgigb45S3DzN/JOVWlppWTadwiHgG/S8hZRyjYUSbCRa5rIk8+Xx4wW527OxYt7FFb6pMnjV3nXf+oRuH+pdrbm8lPg5RO2Ul5jxxlKi6TQWIFx42vYVYgWw6TvBLIubKBPL1f4OYF6vSY1s6M6yK/8xtsPupFLdxXApfOUX87SpdcEaifldy3yQl2naArCunghjihIbGA48an8GKBjubITjy9cqXvLgk8uIy8vjFqLcypAI+3k4Yv1QCAlIWWAgDsZyL5+g/a6ctQ99rSzP64WTA8/91C/6+JXAFXzyIxgULoO8RKSU7nyHaFdtWlHy9XXt9AnlwZ9ULQu1rQ3Sfgix2aJEmsjJSumBQhBMrKJNtqScti17mUqHDvEu3+pdr0foFRnXCBaHhCBKb8Su5oiVcHiREcOn6B5chmEphiVqCRhpWkxYMC3+2mdywmqjXLkAlYQIbXeigPLqcnNQkA2CWSBLg6cvjbeDtQnlKCNbNybRHtMkM9cgZWXx04H1NIR+aLnbRZZWhfHS8REiMohKlPgisleYTsDPj+ssDek7T3LHXHcTPVYULFXkuS1DtX6p0rjVxpNdeMVJawSinmIuC6aPS7ilL5Ur7Wf7b6j/byB72VKmmxXgV/8MpGOqINTmVI7ODoSWVYyfjEV0ryCJkBmNlfGdxQ7jZDffe3MIYht8+4jLHjL5ynfLSN2FWhSWdWihrJftW9f+9JGDg79NlOuuRy+frG4IUgpZdZWkgPnYLL6/trbCP2gqMnNeGVkiRJcqVSkkeQAB5sK8+7NPD8OnLtfK3o9NnjkSSQUSMdnu+m3P6jZtGtGnv3ymrkaSJNWEfbf6n2qK0svCzQpIoSW5AS/CSWL68n950jYdUlJB58Oj+mMJ7Nke0i7apLP18VaFwZ2n6ufvPHWc0w6p/IjU3lelnwQn6Ccs1QgOk7SOvP1Hl7yILLAv/qJLN1MRaDlNG6XiFVlHLXCTp/N/lrM5RBJC5wuXHqwByhAMDnR4STocDYc+W+uXToj+TaxtIznZXM8q7Qa92VLjPUaxo5nqxk8T760HKNAvy3l3JBTtSfxcUSSiVTd/eNmsfMSn7c+ECQLAupXttAbmkuV8YtJUh84HSZ9PBKSWw5DFqBRvhymIvrSGsGBwpLoOknoRfzSYlq9l8NK0n/aK/cudjBZNzbjtG/fE9uXKDd31pefmUgBhW0gmTAPEgJhm2U3gxSnlDh7S3kvnNwtCPxgmMoueGVklACw8LnbjbpS5JUIx3e7618d2lgSSFt+qn66gZyKvL60OFt5KNn4N0t9jtI1xyid/2kdZ+pdaoBvw4J3NjUnSCX0fUaVim9GaR8dwvplSs3rJQcxiviZXDeTFZ0lZJQAnXoNkXo/tqmmvRZP2XWAOX7PbTZp+rkjeRMOLFTJHjrAuXxn7XCEnt6VRyC/2wm3WaoV36n5VaU1g8O/KO9p/OheDZISQFe3UBGtMZhj9iAh7+CSAQ0TSOESJLEN8V7xFXlEfhaGFmW2YWK9M4ONaT/XaysPEhH/aJNWEduaQZX1odONfXvuaWZPGKZNq1PXPsvfzlI//0rmb6D9MqRR3VSBtSVZClFbpwYpIRwjx3lBil1/85un/mHzi6gWUHo6YwzGfEbKITJBJNA5gVNluUMiURcDmr9+pybLX09IPDLQfrRNjJkPpEkdXBD6aqGcvdaZxflj+qktP1CnV1AL8qJrj8lKvx8kC7ZT6fvIEdOw9AW8oZrgrkVozyr5Ed3L8TAJLtfoiKyZxduzUMEaXx5vYab6BG7QCFMDpiLSZIklEAj4qbAmF3EnbOlTjXkcZ3phmPKlzvJPT9p+0volQ3kvnlSo0rS6HPlu5doqwZBtfIWKO4+QZcU0iX76dJCml9E21WXuteSxndR+ubhXrcw6KzJsEd0KkgpXX0INh2BIQ3/tCx1pr9FsxJBGCiEnoYHWmRZTu3soLGh2xdvS5vtq0vtqyujOsH24/SLnXT6DrrjONlxnBadhq5fQ+tqWqNK0DBLOqXB8RAtVqE4BMfOwLEQLQ5BwQkoUWn32vL5taXnu8qdsyUvx/+SBaNB+ewa7ZF2ckZpDnK2XpqblcYwZNj9IaiUCAe/pt6FO0KDQdwnpYfPdDZKoI7GlaSH2/7Z8v4S6PK/UF5FqJspbTtOKwYgKyBlZ0BWEKqkQeWgnBWE2hUAK6Q7zepDdOVB+nHfP0O2JkFKo+uVEX+QEkklUAg9h7gvHh2hRmxxhMZA7Qrw7gXw1x/p2sFK9fSEfSyi5+lV5NF2coa1pUtGR6t4HCIEKflP/majNOIXM5XAaLOHoKU5ssEHlZJigO+IMG5uSww9a8OQRtK9S6wWpkBsZ/UhuuIAub2FbROXxf0hOisTvLGTErELFEJPsHz58vz8fDbL+6dSknVECXTOF2qFcV2U/CL6yfYE5SBFdDy9ijzeXklw5NWYbcALOykRG0Eh9ARbt24FALYixu2+eAvdchjXTeR0Bd7tpdy/VNt1AmexRLPGbnPQLnRmZbmVtiBCgh7USLfw3JDyJzfeeGPbtm1dn+U9hXesQB2dsqW7Wym3LXIwBykSlqdXkcfaJdocjBOj61VnVhr9/N5JYucfUAgRz8G/7a4EAq3wRAf5WAj+/Ss6SBPHmkN0+QFyR8tUm7JiCFICgKiOol66cw7JT6qNKiSp4d/tBK8IjZaADP+9UHlypfbbUZx6EkQymoN2EdagFMOTnn1kTBa8O9cgvkK3ItTt7pRPy6rSyI7KLQs19JAmgFQ1B+NH3P6IWhgzOLAQl+ESyCslud0jq9zXWq4YgInr0EHqOH42B5EEgEKIuIZuOYzb3YkaCeCdXsrL67XlhWgVOsjCvXRNEUVzEHEOHFuICySvFaijXqb0nwsC136vHTzldldSFELhoeXaxK6ertqIJDsohEhC0Umg292xgcvrS9c2km5ZqKJV6ARvbSYZClzTCGcqxEFweCGJgC/yhhSSQM7YLsrxEExYi8FCmzkWgmdWkVe6Kyk1XBDvgUKIOItuR0SKSSAjIMO0PsorG7Qf96FZaCejftEury91zk7BMYN4ChRCxCn8IIGcupnSB70D132v7T3pdldSha3H6PtbydOdMekg4jgohIj9+EoCOX3zpKEtpBt/UHFnoS0MX6r9o72SU8HtfiA+AIUQsRPXKyW5y6hOSkCGZ1djnaZ4mbebbjkK95yDExSSCHCcIbbhzRzZiUSW4IPegamb6dzdaBXGjkpgxDLt5e5KOrpFkYSAQojEC8+R7WcJ5NSqAB/0Vv62UN2NdZpi5bWNJLciXFbP1wMJSSQohEjseLZSkrv0ypXua638Zb52Cl2k0bP7BB27Rnu1OxqDSOJAIURiwfuVktzl8fZy48rSXxfgupmouWOxdl9rpVVVHFRI4kAhRKIjWSoluYsE8J8LlN0n6ahf0CqMgrd+JXtPwuPtcVwhCQUHHGIVMTsaSmC5ZCgw4+LAtG30/a2YccYSvxfTf67U3uutBHFwIYkFRxxSPsleJsItsjPgq/7Kw8u1H/aii7QcKMAdP2qPtlPaVMMBhiQaFELEDFwOEyetqkrv9w5c/72KtezNeX0DOR6CB9vijIS4AA47JDwpUynJdfrXkUafq1zxnXb4tNtd8Srbj9Pn1mj/7aVgdm3EFVAIET2pVynJdYa2kAfWla76Tj2D4UIDhMLfFmpPdlCaV8HBhrgDCiFyFrYdAiXQIV7oplRLl+5ajItI9byQTwIS3Nsa5yLENXDwIX/uiACUQMeQJfigt7LqEB29Bq3CP9lwmE5cp719IfpEETdBIfQ1/iwT4RZZQfhmQOC938gL+aiFAABHzsDgedoL5ykNK+HAQ9wk4HYHEHdg9h9PEOp2dzwHIYQQYvta2dyK8P2lSq9ZGqHwSDtfP4YSCjf+oA6qL93c1NfXAfECKIS+g1mBbH7HffFGNE1j14ddHB43ZcfZe/h1EzXSomTWyZR+uEzp/bWWGYC7Wvn3EeTRFdoZAuO6YE5RxH1QCH0EswIBU8NEgBBy5swZWZYDgQC/PrIsa5pGKQ0EAoqigGBM636KL5gi8p/irwBQj2nhLE0C6c5WfrwRH2wlX+ykK64MBPx49ojnQCH0BbxMEgBIksTna4TBrD1ZljMyMtj14WtoNU1jzw18PVFYK1BnC7J/h7JrccXLnpchze4P/WYTGejtLWUoq52pzepD8NBybf6lgewMt7uCIACAQpjyGCsFogqK8FhgMBjUHaeUKoqSlpbGD4a1BUXNA4MtyCxLJq6iyBFCmlam8wbCxd9ChQC5oTGEbYGha9ORC5Eo9pXAkB/grZ6YSg3xECiEKQufoNELaoRZacwKVBSFqwulVNM0iOA9LleKjBrJbUHREOfLdJtXlecOpBfPoUFFvr6JbGxEPMJNUojsdPU4IQLXzlf/3hyuaIBjEvEQKIQpCJ9JUQLDwmJ+OiuQS6OiKDFfN3NZ4o8m4q/NKtGZF8GlczWiaf/XOLwVyM1K/o+6n0xuCSGqqgYCAZ1A6lTTRe5crOVUkB5vhz4JxFugEKYUohXo+qznNSJZgewgAMiyrHOQ2vjRJhUcO9aE7y6ll3+r7Twh/bOD3i7UGYhhRQ5Kn3syMjL4Yldz5y1vLexPJ3hkuba2iC68PCBR3EaJeAsUwhTBGAtEOKIEilInSqBD1rPF+9KmmrTsysAVc9Wtx8mUnkqaHF6QwgYpuYHLFrjyaiFQNkipa0TXYCS55b9CHEpJAR5cpi3aR+deEsgMQCgUbQMI4iwohEmPubXhcyiloVCIXRydI9QkFmjXR0d1X3IqwKLLA39bqPWdpf7v4vArKnXrfpnmBYNB/hEmQUpjI2DqemUhSWZoih8Xdk2Q2DH9AlqAB5Zpi/fR7y4JVE+3dOkQJMGgECYxaAWawE0l7iXmTkJuBbJ9gU58dGyPJhkKfNRXeXqV1mOmOqu/0ixCNQa+0tWo4laMNp3LNKzXlP/KLpHRDQsR5JaL9Nk3S9IDy+myQvh2gFwtjQLgKEW8CAphUqKbasXnfSSSw5PpIqWUBwiZnOisJaNNY/Ehg4tKPNa5BDCqk1I3k1z4tfppv8AFOWU+mp+aGOOM+iMii6VoBYqbbUwWrIZNssP+kQLct0RbfYjOHShXChBNO9t/8QonMkiJIJFAIUwm+DM4WoFhiSSBkfTDOPPqVJD9yjOrQTilFN9g1035ewu5QZY0ZL760nnKDU1kEBK/xSOBJnCdM19mZQxS8nPXKSVI0v3LaP5hmD1AqZwm8avEF+XqQpWRgpS6S41jHnECFMLkwBZrI4XhMT+dTkSrH2FtQZ008pgZCKYSm6PDmuZhrcxyubiO9N0lgUFztbWHyMh2tEJQZvsiomrEClENLYuu15BG71hMdh6nsy6WKyokFCKapkmlFU6MUcZy94cwwhqmIDy+oEwisYFC6HXEeQq/50ZMHKFsmai9+iGuCqFly1fRsrvmxR5CuOUqUJ5GtqlKl10O9yyFHt9I7/eW21e3+e47FGPefRKu/4FUTYNZAwIZMiGEKooSDAZ1F8ckSBnJag+rlOwBiP00XmR0vSJWQCF0jd9//71BgwYmb0AJNCeSw5OvJXHUhDLqR6R5Vpy+dWLA+qlzvTK7h6l47YryFxfLn27X+s9W72olj+yo2FXB1okiUwAwfw+9ZaH29xbSE+1AAg2gTNYC4+XSYbQCIwUp+Qt+93UNlrsmqFy5RfwDCqE7HDt27LnnnnvrrbfC/pUvw0MJDEtYCaQR9svb/tExe6fNo+XPLwAAIABJREFU51+GqqrsBdMPdlJDGkrn11LuWKydP5P+90K5RdnVpFG5Xo1BOLvQKDy7WntnC53WS+pek8iSLMtR34VypYhJGl/lJD5GhPWaSgJiIxBZbimlLEEPs/V1XcLvY0qCQugOo0eP3rdvn/E4WoHmaJp25swZRVHYmgt+ibgj1LnUMOyFQ/dFVHHxvPiLOlnw9UD5rV/JBV9rD7WRH2kny6W9sOh6FQ1Z2/tfWAI3LlA1QhdfCnmZkqI4excCAbOJS7T/wtqCDFHh+JMN08L09HQeCeYaqetDWKcrfmeTFBRCF1i2bNnMmTObNGnCj4RCocOHD4sSiN8oHVzq+OY/dkS0AtmSGaOKxHMxHdUPiJz1xogEcEdLuXeu9NeF2vd7tcnnK00r6yNnYV2vPH7Gpnt28GybMS3k0fHtLnLrIu3ultLj7ZWA4mB2ArCWO9DkZhmtQC6WbDixZywQrpKolDqz0rrcMoyNIB4B1x+6gKIo7733HntNKX3jjTeaN2++bNkylMCwaJoWCoUAIBAIsOq43ByUJCkYDFasWDE9PZ39ScyxommapmmqqoZCoVAopKoqO0IE+BRmhE9wDiVuZS44ptzBYNDi7v7mVaTFlwf65ck9vlJvW6RtPx5meYgI0zlFUXi1YfFcSGm2bvGC8Muim9+N7DxOblkQGrpIm9ZbfrJTwAkVpEIChPjXS4tXRjS+JUlKS0urWLFiWloa86tLQllKPpb4QFJVlYctWTts+AWDQXYrOexz2bUNOxrFMWkyGhFHQYvQBbp06cL9opIkFRQUfPzxx+PHj3e3V15DNJXEZS9UyI7GTahIz+Bia7qfDBIubRj/lc1x8dtMxs6w3f2xVboIyPBYe/nuc+TJG8l5M9SBdeWRHWVdGhpuyIZNPaP7VWfoiPtDjFrI3lx4krywnry9BYa2kNYPCVQJ2j99RzoFuwi73th4QYy9grLWJImQVUf8VfeIFrYpsFZpCx+UnQCF0H3GjBnjdhc8R9iYXzz6YXGCE/cFUiGftW6CC/vTCvzBP55iT4xKwT/l8PyZ6sC68lOd5KaV/9zXEZt3odxp98gpbfJG8tJGuLqBlH+NklMBKCWUxrI/JBJxnoKV9mPONGvljosCyS1L9onGJ7awAyms01W8wnxk8qaYD9w8eopEAq8a4iG4X05nBdqoH+afbh7F0U1P4rO8cX7UTXCR9jvGCZPDO1rKL67XzpuhDm4o3dFS7pTtiH4Uh+ir67WXN9Ar6kurrlbqZerFklrbH2KURrEF56xAUZ8SkGw9rCaJF0encKItqDMHxdfMxcq+IPwf9+3b9+KLL37zzTf5+fnl+thLSko2bdpUv3797Ozs+M83NUAhTBCEEFVV09LS3O6IR9E5QvnxBEigdfvDxBowOl25iclmLgBg0s7nYt5m/KJVNY0+3VEa0Trw2kZy0wJSosEV9aWrGsq9cqRA3Nfs4Cn6TQH56nc6fw/tX0daPCjQPEI2cPOnAYZOBWnpFghRAsXLCDat6OG7Sx2SQK5n5ipuxYcfSSz5KbAoJntdUFDwwgsvTJ069ZJLLpk1a1a5KrhkyZLhw4e3adNm586d8+bNcyjvfNKBi2UcR1XV9957r2vXrkuWLHG7L16EUspWHwCAuGaEHWcL+cRKQ/Z+NLcP4hQk9u+yAN/LKMtyenp6RkYG35omzmt8CQZfhcFXT+jmxHJPoUaGNKqT8utfAj9cpjStLP3rFy3nw9C187X3fiPHoi8BuOM4fWU9ueibUNNP1U+3k4vrSJv+EvykX0QVtIhkgB3n601A2MwAgpNAt75JZ5ebwJ5B+YpQ2wcS76EkpBmKGeMo4gNJUZSMjAy+LmzHjh133HFH586df/vtt5YtWy5durRFixbXXHONSeN79uy5++67Z8yY8c477+Tl5a1fvz6erqYSaBE6yJkzZz7++ONXXnmlZ8+eX3/9dU5ODv9TpUqVbrnlFhf75gVo6WI8/m0XrSgo3S/vUJTIORcclJfm27xXUNaJRw2xN24h0dIVrbpmG1eShreRhreR95ykX/1Op20j9yzR6mdKdTOhXpZUL1NqkAV1M6W6mVC7gnQ8RPechAOn6L6TsLcEDpTQPSdh4xF6PASX1aUjzpH61VUqBJy6BWAwoXRGpNF8NCqfaGGL/8UCaVBqizt6Ck60DxEyBf76669jxoyZM2fOXXfdtWXLlurVq/P3FxcXm7T25ptvPvLII3l5eQDQsWPH/Pz89u3bO9HtpAOF0CmmTp362muvXXPNNfPnz69atarur5mZmUOGDHGlY16ACsteWKiD+8d4dma5bD1Yhjg/xuAu4zKTeAm0QrnzKRdILgbi6h4weN5yMmBYS2lYS6VEk3YcpwUnoKCY7jpBF+yFXSdIwQkoLKGV06S8ilAzQ8qpCDkVoEVV6cLapFEWbV9DCije0o9IrmljkJLrBx9IYgTOOHJiGEhOSyB3hIrrxfLz8ydOnDh37tw777xzy5YtxoklKyuLvSguLuavOd9///1jjz3GXteuXfvQoUNO9DwZQSF0ilatWi1evNg4Fn1O2JifJKwlYdmZ+XEoawCJ7UgWCiTxIxZDOHGeGlNx5yolsVMI275JkJJSGgRongUtKolLMCTJYHVpmkaIxnxynpJAi/CBxPTDxKzU/RrtQNL5dW2ERsgUuGbNmjFjxixatIhJYOXKlU0aKSgomDNnzu233647fvLkyczMTPb6+PHj/DWCQugUPXr0cLsL3iKSqWTFhApr7oBhggu7AY4IG5/5XMYbicGsNOJQpQuOFf2IZC2Jjeh+EiFtGJ98ddln7Jru3dIPI2FtQSsDiZbmOBUNUN6IXQOJtSNagYsXLx4/fnx+fv4DDzzw7rvvVqhQodx2HnroIeP8c+bMmYyMDP7rb7/9dvXVV8fZ4ZQBhRBxnEg7B8SsafFPIsbYEp+nZKEMnq5jYNgqbmzTBC9IoEXCKiV3tPK8M2KQUlTNSE8h5QowxJep3Ar2Zpo1H0jGRa0QR6UtRiQVX7x48ahRo7Zv3z58+PDPPvssPT3dSv/nzJkzd+5coxDKssxTugPAypUrn3jiCSsN+gEUQsQpeEALElUsUPfRUNYRaiW2pHtDpFUYrH22zMehUwCX9CPSuRidrjqlBEMalEjBS7ugEXIP2fsR3JCFaIKU/E/mPnwqrDgV78K8efOefPLJ4uLiRx999IYbbohqm/zPP//81FNPGY8HAgFVVdlnLV++vHbt2riPkINCiDiCbs8TOyh+7Z2budiLqPQjqgmOV70PBoOi5WQeW4rhFBLgRbR+F4wWoa5N8QW/y1wDxLwqOl0J26D1U3C03ggtzbFgEYs+fKl0RQ8Iz4jsok2fPn3s2LHr1q0bNGjQ0qVLYxgAI0eO/OCDDw4ePGj8U4cOHd5///0KFSqMHTv2o48+irblFAaFELEZ9vUGw+YH0Qp04nP5wzs4YH9IpRvh2cwYthisSWyJNxK2WVFunTsFcFI/dL0Nu2ndPEgptqP7qbvIxkyz9sJvXPybAiNhzBSoquq0adPGjh1bo0aNhx9+WFXVqlWr2v7pEydOvP/+++vVqzdr1qzc3Fx7G09qUAgRexAnWShdv0cFuC5yZ6mNH00TtRDDRMXLtXJMYkt88hUzfcRgTUaClu5XcVE/rAi8+PRgfJLg+sFKbpHSZJ7lNhvVKTjnjg4bLGe7jZ977rlatWpNmDBh0KBBtn/u5MmTO3To0KNHjxo1anz44Ye2t58CoBAiNhA25sf1Q1fXhv8pUgzJPB4jIk6+TpyXvfoRySJkH8RTRzKs7A8J22akU0hAjrr42w/7KENK4QmGqLUgJT8I1gaSc89SYSXw9OnT77777rPPPtuyZct33323e/futn8u+5Rp06b16dPHicZTBhRCJC54LDBsjmwrCzF0cxkIfjOTCY5LgtMSmJgcp8wQtBJb0v1qopQ8OMdXhDp6Cs7pBy1NvMCPlzuQdI8U/IjR6WocY/bCfblisLy4uHjq1KkTJ07s2LHjF1980aVLF9s/l5Oenv7jjz865ONNGVAIkRgJu/khBv0o94FdN8FxO0BccRA2qhTblz/BEmi9k2FtQZ00siAlN6G4foheRF0jsTlgE+CONuqHOdEOJP4MB4JpbvRGxDyWwlqBx44de+ONN55//vnzzjtv5syZHTt2jLZZK+Tm5lasWJH/iipYLiiESNRwb6GxXq4T+qGbqngMkr/BqJSR9nWZT3Ci/eGcBLIXTrQvPhnoloNS0/0hxgsFkTXSlUCaLRgHkjHiq3veEoeWudPVOJCgrIofPHjw9ddfnzx58sCBAxcvXtyiRQuTrp46derLL7+8/vrrYzvTfv36xfaPvgWFEImCsFagczMXx1w/LAaBIPIEx92wXAJFW8GuU6DCjjTbMb8Lkc5F52o2D1Kyg7oEPfZeH0cHEr/pJipuctP5xRFtSt2foPTZgpfcopTu3bv3+eeff//996+77rqVK1fWr1/fpJPFxcVTpkx57733rr/+elVVsdBuYsCrjJQPjbBmJOyTr+0fbYt+RJrgSGmCUHHypeGClBDrBjhHTSjWuPEUosXE3BEfGoyLWs2X81h0vTo9kER3epyXCCLcdDaQKKWiBO7cuXPixInTpk0LhUKSJM2fP59lze7atWukTxk4cOBf/vKXpUuXir5NxGlQCBEzKKUsGwX3I/HpLwFWoNP6wSdf84/QuV7B2gY4EGTAIUOQ34VEJujhmAQpdf+uu0q6j1BV1a40e+an4NCzWtgl01u2bBk7duzXX389bNiw7du3Z2dnHz16dO/evYWFhY0aNTJpraSkZPjw4SUlJbt3765Tp44THUaMoBAi4aHC5ge+Zp3bHyBIoDHyBPF5zEQr0Dn9kCyXibDoehU9ZlwpdadgyyoMSKwJZb19E3PZ6HrlCXr4A5bO12rRmjQ5Bf65iZTA9evXT5gwgVdKqlatGjtepUqVKlWqtGzZstw2H3nkkXnz5lWoUOGKK654/PHHneg5ogOFENEjWnuiI5SWZt8XC9wYJzjeiO5IuROcLf4rc5zTDzbb8v7rVtLyt/HZ2Wgqmesl/3cxnOmQBIID+iGei5jj1GQgGYOUFl2vTg8kGiFH3dq1a1944YXvvvtu2LBhmzdvrlKlikkjhw4dOnz4cNOmTcWDR48eXbdu3U033bRq1SpKaYcOHVAIEwMKIfInkRyeJvoRyayJaoLjB8XJNx5rQEdi9COSCWVUON0/ii90DlhRKcVFudwWt/FcEqMfNEKmWeNAis31SspWSgK7r1LYTOU//fTTuHHj1q1b9+CDD06ZMsW8UhKldMyYMTNnzjx9+vSoUaOuvPJK/qfCwsLbbrvt4YcfZufCktk6ZM4iIiiECICwbUu3ZiRaL6IR8wmOu1t1qzDKjS1Z95vZW+zJSJxRqLCORF37fKUSvzU8oavuEul+WuwPb8QhQxxsyjRr7noVPer8uG5/iMkTmzlUyDQrqjgrFrh+/foRI0ZMnz5dLPgXiYkTJxYWFi5dunTfvn033nijKISnT58ePXo0e11SUuLcGjREBwqhr2FzR1gr0JVKSQwTs4lT7gY4EGogOF2jxwv6IT496J4kTJyu4hscukRh9cPej6CRI8omnglqWPWq+xdxIBnvwrx580aOHHngwIFHH330yy+/tCjwoVBo2rRpy5YtkyQpNzf38OHD4l/btGnDX0+ePHnw4MFW2kTiB4XQv0Sy9vhxV2ausETremVr2Zm0s3fyfYEmbTp6CtG2H61+mPREN/vz9kFQSv7v5s7JaE8hMba4lQcRi2cn+l1Z/9mNZh4LdvDzzz9nlZJee+21O++8M6rHoPz8/M6dO3PDUef8PHTo0IEDB1q0aPHee+/NmDFj7ty51ltG4gGF0I9ECpglxotor37opjPuRdQtVzEGjWLbACc62ZyTQHvvgvG8oOymEdF81D1SQFmzUud0DXuVuH7Ykqk8LLyrttviUqnHnhhyDKmq+tFHH40dO7ZKlSp9+vRp06ZNz549o+3Ali1bxJwybAMo//XIkSNXX321LMu9evWaPXu2FUcrYgsohP5CK4VNsjzUxL2jjq5FhIRs54q0okf3q04ay12FYYzGWQxSWoQmpFIShNOPci0/nVKKBqXoX+X6EQgE2F2w9xJBWSvQiYHE+i/uGgKAUCj00UcfjR49umbNmuPHj4+nUtKuXbt4ZhnmtACAo0ePrlix4qKLLmrSpMnq1atR/xIPCqFfEFe7paWlsYNs8mVfSCZRYSc4k5/lEnYJg43Yoh+6M9JpJJSehW4lLZS1n3SxJevnSxOV5hti1Q8rSskSL/CnkEjWpLHBGAaSQxLIrUA+kFixwGeeeSYnJ+eNN97o27dvnJ+iKIqqqux1fn5+s2bNAODrr78eN27c6tWrA4EAqqAroBCmPjzmp9sUyPUjPT3d5N9FJdCZSlbiLk74r8RTAOez20C4KJTO76rzKJbreuUHvVnpwjri1ppIp2B0utJw+0PCDifx2iZyd+mJEyf+85//PP/88x06dJg2bZpJUrSoaNas2bJly9jrSZMm3XrrrQBw4403XnLJJZhW1EXw0qcskaJN0eqHyWN7pAmOWZ9Qan+IqsnbjGdSpoasN7ZjXT8sPg3ofhXvArs7uuU88fsVXdGPsETreuXHefYZnUyGVc14TkH8Lhw/fvztt98eP358586dZ8yY0alTp3g+Qkfnzp2feeaZoUOHTpky5fjx4xdddBE7Xr16dRs/BYkWFMLUhNkZRivQXv0IO+OzEBEIcTXjBEfC5bMGCxOcOHM5GkgDW/VDPB1+CmK9XCtBSmObkZTSaXd0WP2IB5OBJJWuXhF/0lgrbYmNG09BrJT0/fffl5sOLQZyc3OHDRs2YsSIiy66aMyYMQ7dICRaUAhTikhWoFv6Ea0pYDLB0dIEb+zUHDoFLi2OehGN+mEepJQiJLEL64BlmuqECkbSD3s/AkpPWTwFKzfFog+ff0HEUygsLJw8efKkSZMuueSSJUuW6DKf2cvQoUOHDh3qXPtIDKAQpgi0vEpJTgfSYtMPK//F/GO0NLUKP9Ow/25uCpR7Cg5N7vHoR6Rz4cepUA+EB9XCamQ8rlfevmcrJZXrwyeE8Kr3UPqV+eOPP55//vl33nnn1KlTXbt2rVSp0syZMx944IF4zgVJOlAIkx5CSCgUYjNsPLHAGEiYfkTy5RqdrnzKM7pew1pdohfROSvQaf0Iexd0ZqWV/SFhldJ6LDCeUwCHt2ayjxDTfG/dunXs2LEzZsy44447Vq5cefTo0cLCwt27d4dtpKio6IcffmjYsGHnzp1t7yHiOiiESQy3jcSvNzvCvYhiiAXs8/g5KoFgWT+idb2CsKyfveCnEDZUGf8pgDf0w8T1ylvTCScphd0F2VByK/6TcloCIcLu0g0bNowfP37WrFm33377li1byl2osmjRoocffnjQoEHPPvvs3Llza9Wq5URXERdBIUxKIjk8iZBd0zhzgTD1cCL53MISv/+qXOzVD6NS8lMQF2LowktQNrZkbMpcgPlaRy9IoEV07fBAmm7dk+49kRop1++aMCtQKpspcN26dc8//zyrlLR9+3bzSkmMY8eO3Xfffd9++21OTk4wGJw/f/71119ve4cRd0EhTDJMJDCsfpjoHI2mlKAYCHRIAhOgHyAsxODHTeZio9M1rFOR66LO/rD9LBJgQnErUJfjNKqBFGkhjziQmKvWibMIawUuWbJk7NixK1euHDFixJtvvlmxYkWLrb377ru33XZbTk4OALRt25bvAkRSCRTCpIHH/MImCI1BPyKZNboJjj1ZU2E5jL2rMMDVShfmmCuBGHIjhKiqyu+CuPY1ktM1qjPltyDBXkQrGAdSWENcKlssULc8OOw4jGEgUTsqJYn89NNPzz77LHtdrVq1Y8eORfXvSFKAQuh12AwY1gp0Wj+gdLbS6YdugotnA1wk+8MuHNUP1iC/C2lpaZHEUrQmuQBIpknsxMeRBEigUT9shA8GkwRvkTwTYaOSRo0Mq+KLFy/+17/+tXPnzscee8x6pSQdO3bsaNiwIXtdVFRkxZuKJB0ohJ4mksOT64dzmwItTr66Z38Ts4lDSvMa67ZzcW2way4mhqr39mJFPyKZ3SKRgpTipeBnYfwZ5ynEZgVaxOR5SIeJZwIE21p3ocSBxHWOEDJjxoznnnuuqKho/PjxQ4YMiceZTwjh37ItW7Y0b9485qYQz4JC6FHCSiB1vsYb2K0fxkZI2Up7FmNL5m2KGKN3tmOvfhj7qVNBfhAi2NxWnJO6ptyqdBEbYZ+0+O4gdgrspD755JNx48YxH0bDhg2vvfbaOD+aL7oBgEWLFo0dOzbOBhEPgkLoOTRNU1VV0zSmE9w5qfva247T+hFJxc2nb2pY1m+ilEwkmKvWoa2NLupH2Juic73y42F3UvI/gWmO7DjhUm3iCI2zfTaQjJWSxowZk52dPWbMmHgqJemoV6/eH3/80aBBg3Xr1h0/frxVq1Z2tYx4BxRCr0AIOXr0KFtwweL5fHYjpfXWJWF3cySnGUTvMRMdoTaekdh+zPoR1hbUSaMsy8bAGxfLsD63aB2wNLGVLqz3rVwfqWhHckNWlmVWfsvYSDyuV6fDmeKSaV2lpGeffbZ27dqvv/46T2NtFwMGDBg7dmybNm3efvvtjz76yN7GEY8QZvU8knjefPPNxx9/nFJ66tSpWrVq5ebm1q5dm72oWbNmXl4eO5KXl5eZmcn+xWgKGM2Ccic4e/1XRhKpH2EdjFK4/SHGI+IzhNEGNdof9iI6Qp3WD/MEPcaBZO501Q0kh/of6RRYpaSJEye2atXqueee69atm3kLJ0+ezMrKivajKaUTJkzIzMy84YYbsEZEqoJC6Ak2btxYtWrVvLy8M2fOFBYWzp49+7HHHqtUqdINN9xQXFy8d+/e/fv379+/f8+ePQDARJHpIpNJ8VexuKDJBAelOsGnFXtXYdDEVtqLuQWdUupsSh4u1aX5tmu6d0U/osL4dBV2ULGrJOboAZtOKuwpFBcXT506dcKECZ06dRo1apR52jNVVadNm/bqq68+8cQTV199dfxdQlIPFEIvsnTp0o0bN9588828lDzn5MmTe/bs4bp44MCBvXv37tu3j6VJPHDgQFZWlqiLubm5OTk5tWrVqlOnTs2aNWvVqiUJq/ZNJjgwLOW3OMHFP/mWi9O+XPEUyrUyjUakFder01Zggm3xSE9a7EVYn4SV62McSIcOHXrttdcmTZrUp0+fp59+utyI3dq1a//2t79ddtllI0aMyM7Ojv1skZQGhTDVKCoqYrq4Z8+ewsJCLpN79uzZsGEDANSqVatWrVp5eXnM9ZqTk1OzZs06derUqlUrJyenWrVqrB0Tp2ukCY6vr0uABIJn9EMnjSZ+V90bnM5U7p27ENYnYeLD5wNJXFR14MCBSZMmsWKBI0eObNasmZWuTp069cyZM3fddVe054j4ChRCDzFz5szMzMy+ffs61P6HH37Yq1cvRVHCymRhYeG+fftOnjzJZTInJyc3N1f3q5ibik9wbDkPLa2EB+VFlWLQMEetQIf0w6iR/IiYhQBMg5TWEX25zqksOLCumLfMS24xXzT7lIKCghdffHHq1KmyLN97771dunTp3Lkz3+RuztixY3NzcwsKCv6/vTMNaupq4/glyKIiYgMICIpsoijIIlbRShcQLIt1RluXVjq1uKGIMHSgqERIqaSWuq/TjlKcYrGjItYqnY4SaVEIIFIIbRCGRRMjQmQNN+b9cN6eub1ZCJAEKs/vg5O75NxzU3r/93nOsxQWFvr4+HA4nKGl1QOvNiCEo4K+vr4zZ8589tlnFy9efPfdd5WeI5VKuVxuQ0ODq6vr0qVLdTcT5GVFukiTycePHxsYGOBVSRTLY2NjgzyxSC9xUhceU9WqEjGQXuKv68EROiL6oX6REqGJ61VV4QU93IK2UFopUCAQHDhw4Keffvrkk09cXV0fP36M3t4++uijiIgITYaNi4vLzc1NSUnZsGHDrl271qxZExISoov5A/9p4OVoVPDs2TOhUBgTE6PmnISEhI6OjoCAgG+//ZbNZl+/fl0XT20TExMHBwcHBwdVJ3R2dra2tra0tKSmpl66dCk0NNTW1hbLpEgksrCwUGVNWltbW1lZqXIkYuMS6yU2nlTZT8NkxPVD/dsAHkRNETtkiFP1A/9uWrkj6n8LnUogrUCPQCDIzMzMy8vTpFOSVCrNyMgoKCgICwvbu3cv9ZBIJDp58iRSzdDQ0IqKChBCQBEQwlGBnZ1dWlpaamqqqhPq6+urq6sLCwsNDAyio6OTkpLOnj0bHR2txzn+HzMzMzc3NxcXl/nz52dlZfn4+NBOePr0KdbFp0+ftrS0VFZW4s22tjZkRCJrEq9N4kQRc3Nz+T/gxyLWSOomMaQoDMRIdboYAjRpxB+wfiheYlBVOtXcAlZxHa01YlucKoFVVVUcDufmzZtbtmwRCAQWFhYDDhIREbF8+XIul/vOO+9s2LDByckJH/X398e2Y0NDg52dnS5uBPivA0KoD1ALNFow54wZM1xcXDQcoaSkZPny5fhhsW3btq1bt46IECIYDMahQ4eUHrKysrKysvLw8FB6lCRJkUiEQ15bW1sFAkFxcTGyJhsbGwmCQE5XZE1iHyxOrxw/fjwejWoeKXW9IqhagpMCR1unC80ZsNIs9QXCYJBF7AwoxYx0tyKrtMZQRUXFF198cefOnbi4OM07JRUUFMyaNSsuLo4giODgYB6PRxVC/P+IXC6/fPlybm6utu8GeBUAIdQHCQkJUVFRKOcBPfErKystLCw+//xzDUf4888/Fy9ejDcdHByampp0M1nlyOXywsLChw8fTpkyZfXq1Tivf7CMGzcOrSZ6e3srHi0tLa1Wea7xAAANxklEQVStrQ0ICMBrk0KhsLKy8saNG2hxSCgUGhkZ4ZBXpYkiqhYpUS0V9HCnrgsSg6nSqQbqWqY+9UMNGrpeqZskSdI8sZosUmoO9uVSVRx1SqqqqoqLizt37hz1XWdAcnJyUlJS0Gc3N7f6+nrqUZyAtH///kWLFtnb2w9hzsArDwihnrC0tLS0tFRlJw0IevxR9yimGOqUU6dO3bhxY+3atfX19QsXLrx586ail0kgEOTk5GBB8vLyGmzvNz8/Pz8/P4IgZs6cqeociUSCQ17Ru8Xt27fx5tOnT6dMmYLcrTiB0tLSEuklyhVB41DNR5rNRPzbrKQ5XZXKgHYrlSuiVD+0Al5ZRJuKEqt+kZJQ9tKg+BOpUnEul5uamvro0aOdO3fm5eVRy0FoiEAgUJNNGBkZuWzZssLCwlmzZmVlZQ12cGCMAEL438DV1bW2tnb58uV4Dy4UqQdevnx57Nixe/fuoVd1X1/fxMTE77//nnZaaWmpQCCwtrbm8XhCoTA+Pt7Z2VnrkzE3Nzc3N3d3d1d6VC6Xi0QiVGegqKjo4MGDRkZGwcHBOL2yvb2dJpPUTTs7u0mTJuGhqP/KlbUSRKDHuqqWjcNhCFbgEC4hV5sUqGqRkjaI4ph4J1rORO5o4h+ZvHr1KpvN7u3tTU5OXr16Na12j+agkdHnhoaG6dOnE/+8zTAYjKNHjxYVFWVlZc2ZM2do4wNjARDCUQ1+Qvn6+qalpcXGxqL9QqHQ2tpab9Ooq6ubO3cudlgFBwcnJCQoWkVNTU0hISFr167V28QUMTAwQP7SuXPntrS0JCUlxcTEUAMu+vv7abE8f/31V1FREa7OI5PJlIa84mp2uCQ6zfuqdJFSlVtyQEmjSuDo75SkyiikFlvHEU8XL17MyMioqamRyWTnzp374IMPtHXpmpqaoKAgkiR9fX1jYmI+/fRTR0dHDTMOgbEMCOEoIiAggJq3IBAIlixZcvr06fDw8Hnz5kkkksLCQlRc//Dhw2FhYVq8tEgk2rRpk6IAzJgxY9y4cXw+n+Z9cnR0RL1pqDubm5ufPXvG5/NtbGxWrVqlT6lWSlRUlOJOIyOjadOmTZs2TdW3uru7aQmU5eXl169fx5smJibUWB7aj2ZtbY3sHkWnK1YCxahXpWGfOpVAnUb0yJVVmkU1PzMyMqysrDIzM0NCQtra2obs4a+pqXF3dzcwMOjv70d7uru7KysrPT09DQ0NWSxWcHCwdm4GGAOAEI4igoKCqJvOzs6ZmZl458mTJ+Pj4xMTExkMhqura1pamhYvzWQyT5w4gQWgurr6t99+e/LkSWpq6pw5cxTf96dOnfr8+XOaED558mTmzJne3t5NTU2hoaGJiYnvv/++FiepCc+fP+fxeFKp9K233hrCghNBEBMmTHBycqJGHlKRy+VJSUlhYWEymQzXGaitrUXGpUgkEovFTCZTlTU5depUJpOJh6L9Sy3QgzolUVtuKQ3tGSzYEarTiFbi3wV6UKek9PR0a2trDoeDmwUOrZlDaWkpm83u6+vLzs5mMpn29vZ1dXWOjo6bN2+Ojo5GbtKVK1dq6YaAMQFUlvmP0d/fL5fL9RwpU1dXt2fPHmroeWRk5JEjR9B6DIbP58+aNQt97uzsXLp06a1bt/RZ6bi5uTkkJCQ8PFwmk/3yyy+nT59W35pHF7x8+RItUtJqo+PNzs5Oqi7izBArKysU0aNqkZJQZmUqultViSU1qFVHN64ogX19fefOnUtLS3N3d09LS3v99deHc4m7d+9mZmbK5fKUlBR/f3+8MyoqytjYePv27du2bRvmXQBjExBCYGBevnzp5eXF4/Gwp87b27u8vFz9tzgcjo2NzYcffqj7Cf6fqKioDRs2IO+xQCBYv359cXGxjkyfIYM6bSFdfPz48YkTJyoqKubMmTNlyhS0hyAIZE0iXVRMFNGw0xY1xpWa1zF8s5KGXFmlctQpicPheHt77927d8GCBWpGIEkyPz+/ubl5y5YtSr3BMplsxYoVTCYzKSlp3rx5tKMSicTc3FxLdwOMRcA1CgwMg8FYu3btvn37WCyWkZFRTk4Ofh/HyOXyhw8fUh9Spqamvb29+pxnVVUVLlnu7Ow8c+bM2tra0RYuaGxsbG9vjxLa2tra8vLyrl27tmLFCqxMXV1d2HxEWaclJSXYuBSJRGZmZjjGFbcQoS5SIjXCka7YEYr3qFqkVB8aSoMaSUuVQIlEcuLEia+++urNN9+8efPmgL9/R0fHunXrZs+e3d/fz2Kx0tPTFc8xNDTMzs5WteoMKggME7AIAY2QSqUsFuvatWsGBga2tra5ubno6bN58+YXL15cuHCBIIiAgICDBw8i91d3d3dgYOClS5fUlC3VLj09PUFBQVwuF+/hcDh2dnbr16/XzwT0BqpMS3O64lgesVhsaWmJY3loMmlra4tjaNU4XRWVkvi3aiot0yoWi48ePYo6JaWkpLi5uWlyOzt37vT19d24cSNJkosWLbp//77WfikA0AywCAGNMDY2ZrPZbDabtn/Tpk1isRh9Pn/+fFxcnEgkmjp1qkAgSEpK0psKEgRBkiTNq+bg4NDc3Ky3CSjlxo0bhw4dam9v9/Pzi4+P10ooP5PJZDKZqiwtmUyWnZ0tFos9PDxQnYFHjx6VlJTgKgS9vb2qrEm0ifJkFEURm4DIEYoiQlGxntbW1oMHD545c8bFxSU7O3vBggUaBsJ0d3dzudxvvvmGIAhU9I4kSeiUBOgZ+IMDhgV17cfZ2fnq1atdXV3Pnj2zt7fX8+LcpEmT2tvbqXs6OjrMzMz0OQcaTU1NycnJly9ftre3LygoWLNmDZfLVQx02rp1a2lpKap64+3tvWPHjuFc1NDQUGneCKa3t5fWkLK6uvrWrVs4vRKVwVPqdEUrl6amprgezaNHjw4cOHDp0qWwsLCVK1e2trbu3r0bFTSg5cijziS026+pqfHz88N/KuPHj5dKpSCEgJ6BPzhAy0ycOHHIlUiHiY2NTVNTEzZDy8vLP/744xGZCeLKlSvbtm1DsbXh4eFlZWUXLlxQVKmKiorbt293dHQIhcK+vj5dz8rU1HT69Om0iF8qL168oHagrK6u3rdvH4PB8PDwQK1FcKctJPBbtmzh8/k4LUQVycnJhw4dogkhn8+nelA7OzsHW5YPAIYPrBECrw5Xrlz58ccfz58/z2AwxGLx22+/XVJSMoIP1o0bN8bHx3t6eqLNu3fv5uTkHD9+nHaaj48Pj8fT++w0JS4uTiqVJiYm4rRR3GnL0NDQ29t78uTJAw5SVlbm7+8vkUhoL0lHjx41MzNDLwckSS5evPjevXs6uAkAUAdYhMCrQ2RkJI/H8/HxcXJy4vP5HA5nZM0LoVBILWHj6emZnJxMO4ckycbGxiVLlvT09Li7u+/atUt9poHukEqlSvNTFWtVq++0pUhXV1dMTIzSXoATJ06USCToc1FRkdKeJACga0ZXihUADBMWi1VSUvL1118/ePBgxYoVIzuZiRMnUpctTUxMcD0wTGtr67p1665fv15WVhYbGxsbG/vrr7/qd5oEQRD19fW6cyOTJMlisZRWYPfw8Hj48CFBEFKplM1mb926VUdzAAA1gBACrxomJiaOjo5D7magRby8vKhlB+rq6hTFwMbG5siRIygXxd/fPy8vj8Vi6XWWBJGfn79q1SqBQKDmnJ9//jkhISE2Nra4uHiw40+ePFlV5U9PT8+ysrLjx48HBQW999578+fPH+zgADB8QAgBQFcEBgbm5eXhzbt37/r4+NDOoXkj7ezsenp6SJLUx/z+YcKECfn5+WpO4HK5X375ZXh4eGRkJJvNPnXqlLYubWpq+t133xEEcezYse3bt2trWAAYHHIAAHTG6tWrExMTeTxefn6+l5eXWCxG+6uqqvr6+uRyeUVFxdWrV/H5JEnOmzcPFX/RJyRJLly4UNXR8PBwPp+PPnd1dXl5eXV2dg72EsuWLcPfunfvXmhoaE9Pz9BmCwDaBSxCANAhOTk58+fPP3z4cFFR0fnz51GOQXNz86JFiy5evEgQhL29/Z49e3744Yfnz58/efIkMTExIiJCR3Wxh0xTUxNOcpgwYcKSJUsqKysH/JaaVBAzM7PFixdDpgQwSoD0CQAYARoaGnCVmZaWluPHj5eUlDAYjLCwsB07duhfCGUyWUBAwB9//KF4qKenJzg4uKioCO85e/asVCpV0+qhvLw8PT3dycmJw+HgnYGBgQUFBSOVYwoAaoD0CQAYAai11qZNm6ZYu270YGhoSAt2tbCw+Pvvv5WeXFxcnJGR0dfXt3//flrTpejoaD23DwMADQEhBABAHcbGxlKpVCaT4UDc5uZmW1tb2mkPHjyIj49/7bXX0tPTvby8FMdZt26dzucKAEMC1ggBAFBCe3s7TnX38PD4/fff8aH79+/7+fnRzjczMzty5Ehubq5SFQSA0QwIIQAABIPBoLlnd+3a9cYbb6AYgj179uzevRtlvt++fbuxsXH27Nm0EZycnNzd3fU2YQDQIhAsAwCAEiQSSVtbG17LvHPnTlZWVmNjo4uLC5vNdnV1HdHZAYA2ASEEAAAAxjTgGgUAAADGNCCEAAAAwJgGhBAAAAAY04AQAgAAAGMaEEIAAABgTANCCAAAAIxpQAgBAACAMQ0IIQAAADCmASEEAAAAxjQghAAAAMCY5n+PnlLPjgMEgAAAAABJRU5ErkJggg=="},"metadata":{"image/png":{"height":480,"width":600}},"execution_count":1}],"cell_type":"code","source":["ts = range(0, stop=2pi, length=100)\nplot(xs_ys(pringle.(ts))...)"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

Calculus of vector-valued functions

","metadata":{"internals":{"slide_type":"subslide","slide_helper":"subslide_end"},"slideshow":{"slide_type":"slide"},"slide_helper":"slide_end"}}, {"cell_type":"markdown","source":"

For function $f: R^1 \\rightarrow R^1$ derivatives can be taken using automatic differentiation, as implemented in ForwardDiff or by finite differencing. The latter allows the same definition in the vector valued case, so we will use that:

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["D1 (generic function with 2 methods)"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["D1(r::Function, h=1e-4) = t -> (r(t+h) - r(t-h)) / (2h)"],"metadata":{},"execution_count":1}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["3-element Array{Float64,1}:\n 0.9999999983333334\n 0.0 \n 1.0 "]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["r(t) = [sin(t), cos(t), t]\nD1(r)(0)"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

Though in general it can be a bad idea to iterate differencing, here the approximation isn't bad. This tests the maximum difference of the second derivative's error

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["1.7411125000106153e-8"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["ts = range(0, stop=2pi, length=1000)\nD2(f) = D1(D1(f))\nmaximum([norm(D2(r)(t) - [-sin(t), -cos(t), 0]) for t in ts])"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

As with single variable calculus, the derivative may be found using automatic differentiation, implemented in the ForwardDiff package. This is more accurate than the above and nearly as applicable. Here we overload the adjoint function to provide the mathematical notation f':

","metadata":{}}, {"outputs":[],"cell_type":"code","source":["using ForwardDiff\nD(f::Function, n=1) = n > 1 ? D(D(f), n-1) : t -> ForwardDiff.derivative(f, float(t))\nBase.adjoint(f::Function) = D(f)"],"metadata":{},"execution_count":1}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["
\n\n

Again, this is in the MTH229 convenience package. The adjoint function is understood to be a matrix operation and the postfix prime notation is an alias. Redefining it, as above, is generally considered bad form, as it will conflict with usage for arrays of functions. As this is atpyical usage for this task, and the familiar calculus notation is very useful, we do so here.

\n
\n\n
\n"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["note(\"\"\"Again, this is in the `MTH229` convenience package. The adjoint function is understood to be a matrix operation and the postfix prime notation is an alias. Redefining it, as above, is generally considered bad form, as it will conflict with usage for arrays of functions. As this is atpyical usage for this task, and the familiar calculus notation is very useful, we do so here.\n\"\"\")"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

Then we have much improved accuracy:

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["0.0"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["ts = range(0, stop=2pi, length=1000)\nmaximum([norm(r''(t) - [-sin(t), -cos(t), 0]) for t in ts])"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

Visualizing tangent vectors

","metadata":{"internals":{"slide_type":"subslide"},"slideshow":{"slide_type":"subslide"},"slide_helper":"slide_end"}}, {"cell_type":"markdown","source":"

Adding an arrow to the plot is not as easy as it should be. In Plots the quiver function is used to add an arrow, but as of writing, there is no 3D support for this function. In the following, we use a line segment for that case. As well, the quiver function is well suited for adding many arrows, but a bit cumbersome for just one, given data coming from a vector-valued function. This arrow! function makes adding as single arrow fairly easy, by specifing the point to anchor it and the vector representing it.

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["arrow! (generic function with 2 methods)"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["function arrow!(plt::Plots.Plot, p, v; kwargs...)\n if length(p) == 2\n quiver!(plt, xs_ys([p])..., quiver=Tuple(xs_ys([v])); kwargs...)\n elseif length(p) == 3\n # 3d quiver needs support\n # https://github.com/JuliaPlots/Plots.jl/issues/319#issue-159652535\n # headless arrow instead\n plot!(plt, xs_ys(p, p+v)...; kwargs...)\n\tend\nend\narrow!(p,v;kwargs...) = arrow!(Plots.current(), p, v; kwargs...)"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

Here we see how to use arrow! by adding vectors along the pringle space curve:

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":"Plot(...)","image/png":"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"},"metadata":{"image/png":{"height":480,"width":600}},"execution_count":1}],"cell_type":"code","source":["ts = range(0, stop=2pi, length=100)\nplt = plot(xs_ys(pringle.(ts))..., legend=false)\n\nts = range(0, stop=2pi, length=5)\nfor t0 in ts\n p = pringle(t0)\n T = uvec(pringle'(t0))\n arrow!(plt, p, T, color=\"red\")\nend\nplt"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"
Example: tangent, normal, binormal
","metadata":{"internals":{"slide_type":"subslide"},"slideshow":{"slide_type":"subslide"},"slide_helper":"slide_end"}}, {"cell_type":"markdown","source":"

A smooth space curve has a tangent vector, as drawn, but also a normal and binormal vector for each time $t$. These may be difficult to compute by hand, but relatively straight forward numerically, as seen below.

","metadata":{}}, {"cell_type":"markdown","source":"

The (unit) tangent vector is a the unit vector or $f'$, $T=f'/\\|f'\\|$. The (unit) normal vector is the unit vector of $T'$, $N=T'/\\|T'\\|$. Finally, the binormal is a unit vector perpendicular to both, or proportional to $T × N$.

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":"Plot(...)","image/png":"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"},"metadata":{"image/png":{"height":480,"width":600}},"execution_count":1}],"cell_type":"code","source":["r(t) = pringle(t)\na, b = 0, 2pi\nt0 = 1.0\n\nts = range(a, stop=b, length=100)\nplt = plot(xs_ys(r.(ts))...)\n\nT(t) = uvec(r'(t))\nN(t) = uvec(T'(t))\nB(t) = uvec(T(t) × N(t))\n\np = pringle(t0)\narrow!(p, T(t0))\narrow!(p, N(t0))\narrow!(p, B(t0))"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

There is a fact for fixed-length vector functions (that is, $\\|r(t)\\|=c$ for all $t$) that the derivative and function are orthogonal. It should so so then that not only is $B$ orthogonal to $T$ and $N$, but also $N$ is orthogonal to $T$. This might be visible in the picture, here we verify it for each point in ts:

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["4.163336342344337e-16"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["maximum(abs.([T(t) ⋅ N(t) for t in ts]))"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

This is not the case, were unit vectors not involved:

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["3.9994965106955003"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["T1(t) = r'(t)\nN1(t) = T1'(t)\nmaximum(abs.([T1(t) ⋅ N1(t) for t in ts]))"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

Arclength

","metadata":{"internals":{"slide_type":"subslide"},"slideshow":{"slide_type":"subslide"},"slide_helper":"slide_end"}}, {"cell_type":"markdown","source":"

The length of a parameterized curve is given by the formula

","metadata":{}}, {"cell_type":"markdown","source":"\n$$\ns = \\int_a^b \\| r'(t) \\| dt\n$$\n","metadata":{}}, {"cell_type":"markdown","source":"

We can compute this easily in julia. For our pringle we have the length of one turn is found with:

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["10.540734326381326"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["using QuadGK\nr(t) = [cos(t), sin(t), sin(2t)]\nds(t) = norm(r'(t))\nquadgk(ds, 0, 2pi)[1] # first value returned is answer (second is error)"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

This can also be done with with composition ( typed as \\circ[tab]):

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["10.540734326381326"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["quadgk(norm ∘ r', 0, 2pi)[1]"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

We put this into a function for convenient usage:

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["arclength (generic function with 1 method)"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["arclength(r::Function, a::Real, b::Real) = quadgk(norm ∘ r', a, b)[1]"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

Parameterizing by arc length.

","metadata":{"internals":{"slide_type":"subslide"},"slideshow":{"slide_type":"subslide"},"slide_helper":"slide_end"}}, {"cell_type":"markdown","source":"

Let $r(t)$ be a parameterization of a curve. The same space curve can have many different parameterizations, in fact any monotonically increasing function g gives a new parameterization via r(g(t)). Parameterizing by arclength is a useful way to talk about a specific parameter. Basically, this is a parameterization so the that arclength between 0 and $s$ is just $s$. Mathematically, the arclength function is monotonic, so we just need to take its inverse and call that g. Finding the inverse is always possible due to monotonicity, but may not be algebraically possible. However, we have numeric tools to approximate it. For example,

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["s (generic function with 1 method)"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["r(t) = [cos(t), sin(t), sin(2t)] # pringle\ns(t) = arclength(r, 0, t)"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

How to invert? We use the fzero method from Roots and bracketing, making an assumption that t is positive and the arclength is less than 100:

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["g (generic function with 1 method)"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["using Roots\ng(u) = fzero(t -> s(t) - u, (-1, 100))"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

Then the parameterization is found by composition

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["rs (generic function with 1 method)"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["rs(t) = r(g(t))"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

Here automatic differentiation won't work, as it doesn't work over g, so we use D1, defined above, to find the approximate derivative

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["1.9999999911498048"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["quadgk(t -> norm(D1(rs)(t)), 0, 2)[1] # should be around 2"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

Curvature

","metadata":{"internals":{"slide_type":"subslide"},"slideshow":{"slide_type":"subslide"},"slide_helper":"slide_end"}}, {"cell_type":"markdown","source":"

The curvature can be computed different ways. One is that the curvature is the norm of the derivative of the tangent vector when the curve is parameterized by arc length. But more importantly, we have this formula which does not require a special parameterization, but does require a 3D vector-valued function:

","metadata":{}}, {"cell_type":"markdown","source":"\n$$\n\\kappa = \\frac{\\| r'(t) \\times r''(t) \\|}{\\| r'(t) \\|^3}\n$$\n","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["kappa (generic function with 1 method)"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["kappa(t) = norm( r'(t) × r''(t) ) / norm(r'(t))^3"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

Symbolic math and parameterized curves

","metadata":{"internals":{"slide_type":"subslide","slide_helper":"subslide_end"},"slideshow":{"slide_type":"slide"},"slide_helper":"slide_end"}}, {"cell_type":"markdown","source":"

Many of the above computations can be done symbolically to match the work done with paper and pencil. Here we see how:

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["(t,)"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["using SymPy\n@vars t # create a symbolic variable"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

With the function

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["r (generic function with 1 method)"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["r(t) = [sin(t), cos(t), t]"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

We can create a symbolic expression by evaluating r at t:

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/latex":["\\[ \\left[ \\begin{array}{r}\\sin{\\left (t \\right )}\\\\\\cos{\\left (t \\right )}\\\\t\\end{array} \\right] \\]"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["u = r(t)"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

The usual operations work as expected:

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/latex":["\\[ \\left[ \\begin{array}{r}0\\\\0\\\\0\\end{array} \\right] \\]"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["2u\nu + u\nu ⋅ u\nu × u"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

More complicated expressions are possible

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/latex":["\\begin{equation*}t \\overline{t} + \\sin{\\left (t \\right )} \\sin{\\left (\\overline{t} \\right )} + \\cos{\\left (t \\right )} \\cos{\\left (\\overline{t} \\right )} - \\left|{t}\\right|^{2} - \\left|{\\sin{\\left (t \\right )}}\\right|^{2} - \\left|{\\cos{\\left (t \\right )}}\\right|^{2}\\end{equation*}"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["u ⋅ u - norm(u)^2"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

The above should be $0$ – for real valued vectors, but as SymPy does not make that assumption unless asked. Here we try again:

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/latex":["\\begin{equation*}0\\end{equation*}"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["@vars t real=true\nu = r(t)\nu ⋅ u - norm(u)^2"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

We can compute quantities symbolically. For example, the arclength (where we use t twice, though differently):

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/latex":["\\begin{equation*}\\sqrt{2} t\\end{equation*}"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["integrate(norm(diff.(u)), (t, 0, t)) |> simplify # integrate(ex, (var, a, b))"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

And the curvature:

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/latex":["\\begin{equation*}\\frac{1}{2}\\end{equation*}"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["κ = norm(diff.(u) × diff.(u,t,2)) / norm(diff.(u))^3 |> simplify"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

Of course, not all functions are so tractable as this example and numeric integration may still be required to get an answer.

","metadata":{}} ], "metadata": { "language_info": { "file_extension": ".jl", "mimetype": "application/julia", "name": "julia", "version": "0.6" }, "kernelspec": { "display_name": "Julia 1.0.0", "language": "julia", "name": "julia-1.0" } }, "nbformat": 4, "nbformat_minor": 2 }