CIRCLE_SEGMENT
Area, Height, Angle, Sampling and Quadrature

CIRCLE_SEGMENT is a C++ library which carries out various computations associated with a circle segment, using gnuplot to illustrate some of the computations.

Begin with a circle of radius R. Choose two points P1 and P2 on the circle, and draw the chord P1:P2. This chord divides the circle into two pieces, each of which is called a circle segment. Consider one of the pieces. The "angle" THETA of this segment is the angle P1:C:P2, where C is the center of the circle. Let Q be the point on the chord P1:P2 which is closest to C. The "height" H of the segment is the distance from Q to the perimeter of the circle.

This library considers various computations, including:

Determine the angle THETA, given R and H.

Determine the height H, given R and THETA.

Determine the height H, given R and AREA.

Determine the width W, given R and H.

Determine the area, given R and H.

Determine the centroid, given R and H.

Select points uniformly at random from a segment, given R and H.

Determine a cumulative density function (CDF) for the height H2 of a circle segment defined by a point selected at random from a circle segment of height H.

Determine a quadrature rule that can be used to estimate integrals of functions f(x,y) over the segment.

CIRCLE_SEGMENT creates some graphics plots using gnuplot.

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

CIRCLE_SEGMENT is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

CIRCLE_RULE, a C++ library which computes quadrature rules for the unit circle in 2D, that is, the circumference of the circle of radius 1 and center (0,0).

GEOMETRY, a C++ library which performs geometric calculations in 2, 3 and M dimensional space, including the computation of angles, areas, containment, distances, intersections, lengths, and volumes.

GNUPLOT, C++ programs which illustrate how a program can write data and command files so that gnuplot can create plots of the program results.

STROUD, a C++ library which defines quadrature rules for a variety of M-dimensional regions, including the interior of the square, cube and hypercube, the pyramid, cone and ellipse, the hexagon, the M-dimensional octahedron, the circle, sphere and hypersphere, the triangle, tetrahedron and simplex, and the surface of the circle, sphere and hypersphere.

Source Code:

• circle_segment.cpp, the source code.
• circle_segment.hpp, the include file.
• circle_segment.sh, BASH commands to compile the source code.

Examples and Tests:

• circle_segment_prb.cpp, a sample calling program.
• circle_segment_prb.sh, BASH commands to compile and run the sample program.
• circle_segment_prb_output.txt, the output file.

Test number 6 creates some samples of points in circle segments, and writes out data files and gnuplot command files to create graphics:

• sample01_boundary.txt, data defining the boundary of the circle segment.
• sample01_commands.txt, commands to gnuplot to create a plot.
• sample01_data.txt, the sample data.
• sample01.png, the plot
• sample02_boundary.txt, data defining the boundary of the circle segment.
• sample02_commands.txt, commands to gnuplot to create a plot.
• sample02_data.txt, the sample data.
• sample02.png, the plot
• sample03_boundary.txt, data defining the boundary of the circle segment.
• sample03_commands.txt, commands to gnuplot to create a plot.
• sample03_data.txt, the sample data.
• sample03.png, the plot
• sample04_boundary.txt, data defining the boundary of the circle segment.
• sample04_commands.txt, commands to gnuplot to create a plot.
• sample04_data.txt, the sample data.
• sample04.png, the plot

List of Routines:

• CIRCLE_SEGMENT_ANGLE_FROM_CHORD computes the angle of a circle segment.
• CIRCLE_SEGMENT_ANGLE_FROM_CHORD_ANGLES computes angle of a circle segment.
• CIRCLE_SEGMENT_ANGLE_FROM_HEIGHT computes the angle of a circle segment.
• CIRCLE_SEGMENT_AREA_FROM_ANGLE computes the area of a circle segment.
• CIRCLE_SEGMENT_AREA_FROM_CHORD computes the area of a circle segment.
• CIRCLE_SEGMENT_AREA_FROM_HEIGHT computes the area of a circle segment.
• CIRCLE_SEGMENT_AREA_FROM_SAMPLE computes the area of a circle segment.
• CIRCLE_SEGMENT_CDF computes a CDF related to a circle segment.
• CIRCLE_SEGMENT_CENTROID_FROM_CHORD computes the centroid of a circle segment.
• CIRCLE_SEGMENT_CENTROID_FROM_HEIGHT computes centroid of a circle segment.
• CIRCLE_SEGMENT_CENTROID_FROM_SAMPLE estimates a circle segment centroid.
• CIRCLE_SEGMENT_CONTAINS_POINT reports whether a point is in a circle segment.
• CIRCLE_SEGMENT_HEIGHT_FROM_ANGLE: height of a circle segment from angle.
• CIRCLE_SEGMENT_HEIGHT_FROM_AREA: height of a circle segment from area.
• CIRCLE_SEGMENT_HEIGHT_FROM_CHORD: height of a circle segment from chord.
• CIRCLE_SEGMENT_SAMPLE_FROM_CHORD samples points from a circle segment.
• CIRCLE_SEGMENT_SAMPLE_FROM_HEIGHT samples points from a circle segment.
• CIRCLE_SEGMENT_WIDTH_FROM_HEIGHT computes the width of a circle segment.
• FILENAME_INC increments a partially numeric file name.
• R8_ACOS computes the arc cosine function, with argument truncation.
• R8_ASIN computes the arc sine function, with argument truncation.
• R8_ATAN computes the inverse tangent of the ratio Y / X.
• R8_EPSILON returns the R8 round off unit.
• R8_MAX returns the maximum of two R8's.
• R8_MIN returns the minimum of two R8's.
• R8_UNIFORM_01 returns a pseudorandom R8 scaled to [0,1].
• R8MAT_UNIFORM_01_NEW fills an R8MAT with pseudorandom values scaled to [0,1].
• R8VEC_LINSPACE_NEW creates a vector of linearly spaced values.
• R8VEC_SUM returns the sum of an R8VEC.
• R8VEC_UNIFORM_01_NEW returns a unit pseudorandom R8VEC.
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• TRIDISOLVE solves a tridiagonal system of linear equations.

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