Begin by loading our packages for plotting and zero finding:
","metadata":{}}, {"outputs":[],"cell_type":"code","source":["using Plots\nusing Root\nusing ForwardDiff\nD(f, k=1) = k > 1 ? D(D(f),k-1) : x -> ForwardDiff.derivative(f, float(x))\nusing QuadGK"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"Parametric plots are ones where the points plotted are parametrized by $t$: $(g(t), f(t))$. The plot function will produce parametric plots when called with two function objects, plot(g,f, a, b).
This can be used to plot functions in polar representation by converting to Cartesian coordinates:
","metadata":{}}, {"cell_type":"markdown","source":"\n$$\nx(\\theta) = r(\\theta) \\cdot \\cos(\\theta), \\quad\ny(\\theta) = r(\\theta) \\cdot \\sin(\\theta)\n$$\n","metadata":{}}, {"cell_type":"markdown","source":"but it is easier to let the graphing function do that work. Here is a specialized function:
","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["polar (generic function with 1 method)"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["function polar(r, a, b)\n\t plot(t -> r(t)*cos(t), t->r(t)*sin(t), a, b)\nend"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"Which is used as follows:
","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(theta) = 1 - cos(theta)\t# a cardiod\npolar(r, 0, 2pi)\t\t"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"If $c(t) = (x(t), y(t))$ parametrically describes a curve with differentiable functions, then the tangent line $t$ is given by: $dy/dx = y'(t)/x'(t)$. Using D we can compute this quickly.
For example, the curve $x(t) = \\cos^3(t)$, $y(t) = \\sin^3(t)$ is an astroid. Its tangent line at $\\theta = \\pi/3$ is found with
","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["((0.12500000000000008, 0.6495190528383289), -1.7320508075688767)"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["x(t) = cos(t)^3\ny(t) = sin(t)^3\nt0 = pi/3\nm = D(y)(t0) / D(x)(t0)\n(x(t0), y(t0)), m\t\t# return point and slope for y = y0 + m (x-x0)"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"The area bounded by a curve in polar form given by $r(\\theta)$ between the rays $\\theta=a$ and $\\theta=b$ is
","metadata":{}}, {"cell_type":"markdown","source":"\n$$\nArea = \\frac{1}{2}\\int_a^b (r(\\theta))^2 d\\theta.\n$$\n","metadata":{}}, {"cell_type":"markdown","source":"We can check this works for a known figure. When $r=1$, $a=0$, and $b=2\\pi$ the area is that of a circle of radius 1, or $\\pi$. Here we integrate to compare:
","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["0.0"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["r(theta) = 1\na, b = 0, 2pi\na, err = quadgk(t -> (1/2)*r(t)^2, a, b)\na - pi"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"A bullet fired from a gun follows a trajectory given by
\nLet $a=500$ and $b=300$. Show graphically or analytically that the bullet leaves the gun at an angle $\\theta=\\tan^{-1}(b/a)$ and lands at a distance $ab/16$ from the origin.
","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"A cycloid is the shape made by tracking a fixed point along a wheel as it moves away in a direction. If the wheel has radius $R$ and the point is at the bottom of the wheel at $t=0$ then these functions parameterize the curve:
\nVerify the following property algebraically. The tangent line at time $t$ wil go through the top point of the circle of radius $R$ with center at $(R\\cdot t, R)$, that is, it contains the point $(R\\cdot t, 2R)$.
","metadata":{}}, {"cell_type":"markdown","source":"Assuming $R=1$, prove this analytically.
","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"A Bezier Curve is a parametric curve which has been found useful in computer graphics. The cubic case is described on Wikipedia by
\n\n","metadata":{}}, {"cell_type":"markdown","source":"Four points P0, P1, P2 and P3 in the plane or in higher-dimensional
\n
space define a cubic Bézier curve. The curve starts at P0 going toward P1 and arrives at P3 coming from the direction of P2. Usually, it will not pass through P1 or P2; these points are only there to provide directional information. The distance between P0 and P1 determines \"how long\" the curve moves into direction P2 before turning towards P3.
","metadata":{}}, {"cell_type":"markdown","source":"If we write $p_i = (a_i, b_i)$ for $i = 0, 1, 2, 3$ then the parametric curve is given by:
","metadata":{}}, {"cell_type":"markdown","source":"\n$$\nx(t) = a_0(1-t)^3 + 3a_1t(1-t)^2 + 3a_2t^2(1-t) + a_3t^3, \\quad\ny(t) = b_0(1-t)^3 + 3b_1t(1-t)^2 + 3b_2t^2(1-t) + b_3t^3\n$$\n","metadata":{}}, {"cell_type":"markdown","source":"where $t$ is in $[0,1]$.
","metadata":{}}, {"cell_type":"markdown","source":"If we input the points as pairs like (a,b) into julia we can construct and plot the functions via:
Find a group of four points where the graph is convex (always concave up or down). Find a group of four points where the graph is not convex (it has part of it concave up and some concave down).
","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"The tractrix with parameter $l > 0$ is described parametrically by
\nAt a given value $t$, the tangent line to the curve intersects the $x$-axis. Show that the distance of this line segment is $l$ for t = rand() (that is a random value of $t$ in $(0,1)$).
Plot the polar curve described by
\nfor $a=1$, $b=2$. What is the shape of the curve?
","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"Plot the polar curve given by $r = \\sin(\\theta) + \\cos(3\\theta)$. What more closely describes the shape: a circle, an ellipse, or something odder.
\nThe polar curve given by $r(\\theta) = \\theta^2 + 4\\theta$ intersects the $y$-axis when $\\theta = 0$ and When $\\theta = \\pi/2$. Find the bounded area between the curve and the $y$-axis.
\nThe cardiod $r(\\theta) = 1 - \\cos(\\theta)$ has some of its area to the right of the $y$ axis. The total area is given by:
\nWhat percent of the area is to the right of the $y$ axis?
","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"Find the area of one loop of the lemniscate with equation $r = \\cos(2\\theta)$. (Finding the right limits is all the battle.)
\nPlotting the polar curve $r(\\theta) = \\theta \\cdot\\sin(\\theta)$ over $[0, k\\pi]$ produces $k$ nested shapes ($k$ is a positive integer). Find the area of the smallest one.
\n