CIRCLE_SEGMENT
Area, Height, Angle, Sampling and Quadrature

CIRCLE_SEGMENT is a MATLAB library which carries out computations associated with a circle segment, including height, width, angle, area, centroid, sampling, and quadrature.

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. The "width" W of the segment is the length of P1:P2.

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.

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 FORTRAN90 version and a MATLAB version.

Related Data and Programs:

CIRCLE_RULE, a MATLAB 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).

circle_segment_test

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

STROUD, a MATLAB 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.

Reference:

• Gaspare da Fies, Marco Vianello,
  Trigonometric gaussian quadrature on subintervals of the period,
  Electronic Transactions on Numerical Analysis,
  Volume 39, pages 102-112, 2012.
• Walter Gautschi,
  Orthogonal Polynomials: Computation and Approximation,
  Oxford, 2004,
  ISBN: 0-19-850672-4,
  LC: QA404.5 G3555.

Source Code:

• circle_segment_angle_from_chord.m, computes the angle of a circle segment from the chord endpoints.
• circle_segment_angle_from_chord_angles.m, computes the angle of a circle segment from the chord angles.
• circle_segment_angle_from_height.m, computes the angle of a circle segment from its height.
• circle_segment_area_from_angle.m, computes the area of a circle segment from the angle.
• circle_segment_area_from_chord.m, computes the area of a circle segment from the coordinates of the endpoints of the chord.
• circle_segment_area_from_height.m, computes the area of a circle segment from the height.
• circle_segment_area_from_sample.m, estimates the area of a circle segment, described by the coordinates of the endpoints of the chords, using Monte Carlo sampling over the entire circle.
• circle_segment_cdf.m, computes a cumulative density function (CDF) for heights H2 of circle segments formed by choosing a point at random in a circle segment of height H.
• circle_segment_centroid_from_chord.m, computes the centroid of a circle segment from the coordinates of the endpoints of the chord.
• circle_segment_centroid_from_height.m, computes the centroid of a circle segment from its height.
• circle_segment_centroid_from_sample.m, computes the centroid of a circle segment from a Monte Carlo sample.
• circle_segment_contains_point.m, determines whether a circle segment contains a point.
• circle_segment_height_from_angle.m, computes the height of a circle segment from its angle.
• circle_segment_height_from_angle.m, computes the height of a circle segment from its angle.
• circle_segment_height_from_area.m, computes the height of a circle segment from its area.
• circle_segment_height_from_chord.m, computes the height of a circle segment from its chord endpoints.
• circle_segment_sample_from_chord.m, computes random sample points from a circle segment based on the chord endpoints.
• circle_segment_sample_from_height.m, computes random sample points from a circle segment based on the height.
• circle_segment_width_from_height.m, computes the width of a circle segment from its height.
• circle01_to_circle.m, maps points from the unit circle to a general one.
• gauss.m, computes a Gauss quadrature rule, by Walter Gautschi.
• gqcircsect.m, computes a Gauss quadrature rule for a circle sector in the unit circle.
• gqcircsegm.m, computes a Gauss quadrature rule for a circle segment in the unit circle.
• r_jacobi.m, computes recurrence coefficients for monic Jacobi polynomials, by Dirk Laurie and Walter Gautschi.
• r8_acos.m, computes the arc cosine.
• r8_asin.m, computes the arc sine.
• r8_atan.m, computes the arc tangent.
• r8_uniform_01.m, returns a random R8 in [0,1].
• r8mat_uniform_01.m, returns a random R8MAT with values in [0,1].
• r8vec_uniform_01.m, returns a random R8VEC with values in [0,1].
• timestamp.m, prints the current YMDHMS date as a time stamp.
• tridisolve.m, solves a tridiagonal system of linear equations.
• trigauss.m, computes a trigonometric gaussian quadrature rule, by Gaspare Da Fies, Alvise Sommariva, Marco Vianello.
• trigauss_conversion.m, converts Gauss to trigonometric Gauss quadrature, by Gaspare Da Fies, Alvise Sommariva, Marco Vianello.
• weights_3t.m, Gauss quadrature weights from 3 term recurrence coefficients, by Gaspare Da Fies, Alvise Sommariva, Marco Vianello.