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Merge pull request #586 from RonaldoCMP/master

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Revar Desmera 2021-06-23 15:37:31 -07:00 committed by GitHub
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@ -10,7 +10,8 @@
// Function: point_on_segment2d() // Function: point_on_segment2d()
// Usage: // Usage:
// point_on_segment2d(point, edge); // pt = point_on_segment2d(point, edge);
// Topics: Geometry, Points, Segments
// Description: // Description:
// Determine if the point is on the line segment between two points. // Determine if the point is on the line segment between two points.
// Returns true if yes, and false if not. // Returns true if yes, and false if not.
@ -45,7 +46,8 @@ function _valid_plane(p, eps=EPSILON) = is_vector(p,4) && ! approx(norm(p),0,eps
// Function: point_left_of_line2d() // Function: point_left_of_line2d()
// Usage: // Usage:
// point_left_of_line2d(point, line); // pt = point_left_of_line2d(point, line);
// Topics: Geometry, Points, Lines
// Description: // Description:
// Return >0 if point is left of the line defined by `line`. // Return >0 if point is left of the line defined by `line`.
// Return =0 if point is on the line. // Return =0 if point is on the line.
@ -60,7 +62,8 @@ function point_left_of_line2d(point, line) =
// Function: collinear() // Function: collinear()
// Usage: // Usage:
// collinear(a, [b, c], [eps]); // test = collinear(a, [b, c], [eps]);
// Topics: Geometry, Points, Collinearity
// Description: // Description:
// Returns true if the points `a`, `b` and `c` are co-linear or if the list of points `a` is collinear. // Returns true if the points `a`, `b` and `c` are co-linear or if the list of points `a` is collinear.
// Arguments: // Arguments:
@ -80,7 +83,8 @@ function collinear(a, b, c, eps=EPSILON) =
// Function: point_line_distance() // Function: point_line_distance()
// Usage: // Usage:
// point_line_distance(line, pt); // pt = point_line_distance(line, pt);
// Topics: Geometry, Points, Lines, Distance
// Description: // Description:
// Finds the perpendicular distance of a point `pt` from the line `line`. // Finds the perpendicular distance of a point `pt` from the line `line`.
// Arguments: // Arguments:
@ -97,6 +101,7 @@ function point_line_distance(pt, line) =
// Function: point_segment_distance() // Function: point_segment_distance()
// Usage: // Usage:
// dist = point_segment_distance(pt, seg); // dist = point_segment_distance(pt, seg);
// Topics: Geometry, Points, Segments, Distance
// Description: // Description:
// Returns the closest distance of the given point to the given line segment. // Returns the closest distance of the given point to the given line segment.
// Arguments: // Arguments:
@ -115,6 +120,8 @@ function point_segment_distance(pt, seg) =
// Function: segment_distance() // Function: segment_distance()
// Usage: // Usage:
// dist = segment_distance(seg1, seg2); // dist = segment_distance(seg1, seg2);
// Topics: Geometry, Segments, Distance
// See Also: convex_collision(), convex_distance()
// Description: // Description:
// Returns the closest distance of the two given line segments. // Returns the closest distance of the two given line segments.
// Arguments: // Arguments:
@ -130,8 +137,9 @@ function segment_distance(seg1, seg2) =
// Function: line_normal() // Function: line_normal()
// Usage: // Usage:
// line_normal([P1,P2]) // vec = line_normal([P1,P2])
// line_normal(p1,p2) // vec = line_normal(p1,p2)
// Topics: Geometry, Lines
// Description: // Description:
// Returns the 2D normal vector to the given 2D line. This is otherwise known as the perpendicular vector counter-clockwise to the given ray. // Returns the 2D normal vector to the given 2D line. This is otherwise known as the perpendicular vector counter-clockwise to the given ray.
// Arguments: // Arguments:
@ -172,7 +180,8 @@ function _general_line_intersection(s1,s2,eps=EPSILON) =
// Function: line_intersection() // Function: line_intersection()
// Usage: // Usage:
// line_intersection(l1, l2); // pt = line_intersection(l1, l2);
// Topics: Geometry, Lines, Intersections
// Description: // Description:
// Returns the 2D intersection point of two unbounded 2D lines. // Returns the 2D intersection point of two unbounded 2D lines.
// Returns `undef` if the lines are parallel. // Returns `undef` if the lines are parallel.
@ -190,7 +199,8 @@ function line_intersection(l1,l2,eps=EPSILON) =
// Function: line_ray_intersection() // Function: line_ray_intersection()
// Usage: // Usage:
// line_ray_intersection(line, ray); // pt = line_ray_intersection(line, ray);
// Topics: Geometry, Lines, Rays, Intersections
// Description: // Description:
// Returns the 2D intersection point of an unbounded 2D line, and a half-bounded 2D ray. // Returns the 2D intersection point of an unbounded 2D line, and a half-bounded 2D ray.
// Returns `undef` if they do not intersect. // Returns `undef` if they do not intersect.
@ -209,7 +219,8 @@ function line_ray_intersection(line,ray,eps=EPSILON) =
// Function: line_segment_intersection() // Function: line_segment_intersection()
// Usage: // Usage:
// line_segment_intersection(line, segment); // pt = line_segment_intersection(line, segment);
// Topics: Geometry, Lines, Segments, Intersections
// Description: // Description:
// Returns the 2D intersection point of an unbounded 2D line, and a bounded 2D line segment. // Returns the 2D intersection point of an unbounded 2D line, and a bounded 2D line segment.
// Returns `undef` if they do not intersect. // Returns `undef` if they do not intersect.
@ -228,7 +239,8 @@ function line_segment_intersection(line,segment,eps=EPSILON) =
// Function: ray_intersection() // Function: ray_intersection()
// Usage: // Usage:
// ray_intersection(s1, s2); // pt = ray_intersection(s1, s2);
// Topics: Geometry, Lines, Rays, Intersections
// Description: // Description:
// Returns the 2D intersection point of two 2D line rays. // Returns the 2D intersection point of two 2D line rays.
// Returns `undef` if they do not intersect. // Returns `undef` if they do not intersect.
@ -247,7 +259,8 @@ function ray_intersection(r1,r2,eps=EPSILON) =
// Function: ray_segment_intersection() // Function: ray_segment_intersection()
// Usage: // Usage:
// ray_segment_intersection(ray, segment); // pt = ray_segment_intersection(ray, segment);
// Topics: Geometry, Rays, Segments, Intersections
// Description: // Description:
// Returns the 2D intersection point of a half-bounded 2D ray, and a bounded 2D line segment. // Returns the 2D intersection point of a half-bounded 2D ray, and a bounded 2D line segment.
// Returns `undef` if they do not intersect. // Returns `undef` if they do not intersect.
@ -266,7 +279,8 @@ function ray_segment_intersection(ray,segment,eps=EPSILON) =
// Function: segment_intersection() // Function: segment_intersection()
// Usage: // Usage:
// segment_intersection(s1, s2); // pt = segment_intersection(s1, s2);
// Topics: Geometry, Segments, Intersections
// Description: // Description:
// Returns the 2D intersection point of two 2D line segments. // Returns the 2D intersection point of two 2D line segments.
// Returns `undef` if they do not intersect. // Returns `undef` if they do not intersect.
@ -285,7 +299,8 @@ function segment_intersection(s1,s2,eps=EPSILON) =
// Function: line_closest_point() // Function: line_closest_point()
// Usage: // Usage:
// line_closest_point(line,pt); // pt = line_closest_point(line,pt);
// Topics: Geometry, Lines, Distance
// Description: // Description:
// Returns the point on the given 2D or 3D `line` that is closest to the given point `pt`. // Returns the point on the given 2D or 3D `line` that is closest to the given point `pt`.
// The `line` and `pt` args should either both be 2D or both 3D. // The `line` and `pt` args should either both be 2D or both 3D.
@ -343,7 +358,8 @@ function line_closest_point(line,pt) =
// Function: ray_closest_point() // Function: ray_closest_point()
// Usage: // Usage:
// ray_closest_point(seg,pt); // pt = ray_closest_point(seg,pt);
// Topics: Geometry, Rays, Distance
// Description: // Description:
// Returns the point on the given 2D or 3D ray `ray` that is closest to the given point `pt`. // Returns the point on the given 2D or 3D ray `ray` that is closest to the given point `pt`.
// The `ray` and `pt` args should either both be 2D or both 3D. // The `ray` and `pt` args should either both be 2D or both 3D.
@ -406,7 +422,8 @@ function ray_closest_point(ray,pt) =
// Function: segment_closest_point() // Function: segment_closest_point()
// Usage: // Usage:
// segment_closest_point(seg,pt); // pt = segment_closest_point(seg,pt);
// Topics: Geometry, Segments, Distance
// Description: // Description:
// Returns the point on the given 2D or 3D line segment `seg` that is closest to the given point `pt`. // Returns the point on the given 2D or 3D line segment `seg` that is closest to the given point `pt`.
// The `seg` and `pt` args should either both be 2D or both 3D. // The `seg` and `pt` args should either both be 2D or both 3D.
@ -463,7 +480,8 @@ function segment_closest_point(seg,pt) =
// Function: line_from_points() // Function: line_from_points()
// Usage: // Usage:
// line_from_points(points, [fast], [eps]); // line = line_from_points(points, [fast], [eps]);
// Topics: Geometry, Lines, Points
// Description: // Description:
// Given a list of 2 or more collinear points, returns a line containing them. // Given a list of 2 or more collinear points, returns a line containing them.
// If `fast` is false and the points are coincident or non-collinear, then `undef` is returned. // If `fast` is false and the points are coincident or non-collinear, then `undef` is returned.
@ -490,6 +508,7 @@ function line_from_points(points, fast=false, eps=EPSILON) =
// Usage: // Usage:
// C = law_of_cosines(a, b, c); // C = law_of_cosines(a, b, c);
// c = law_of_cosines(a, b, C); // c = law_of_cosines(a, b, C);
// Topics: Geometry, Triangles
// Description: // Description:
// Applies the Law of Cosines for an arbitrary triangle. Given three side lengths, returns the // Applies the Law of Cosines for an arbitrary triangle. Given three side lengths, returns the
// angle in degrees for the corner opposite of the third side. Given two side lengths, and the // angle in degrees for the corner opposite of the third side. Given two side lengths, and the
@ -523,6 +542,7 @@ function law_of_cosines(a, b, c, C) =
// Usage: // Usage:
// B = law_of_sines(a, A, b); // B = law_of_sines(a, A, b);
// b = law_of_sines(a, A, B); // b = law_of_sines(a, A, B);
// Topics: Geometry, Triangles
// Description: // Description:
// Applies the Law of Sines for an arbitrary triangle. Given two triangle side lengths and the // Applies the Law of Sines for an arbitrary triangle. Given two triangle side lengths and the
// angle between them, returns the angle of the corner opposite of the second side. Given a side // angle between them, returns the angle of the corner opposite of the second side. Given a side
@ -556,7 +576,8 @@ function law_of_sines(a, A, b, B) =
// Function: tri_calc() // Function: tri_calc()
// Usage: // Usage:
// tri_calc(ang,ang2,adj,opp,hyp); // triangle = tri_calc(ang,ang2,adj,opp,hyp);
// Topics: Geometry, Triangles
// Description: // Description:
// Given a side length and an angle, or two side lengths, calculates the rest of the side lengths // Given a side length and an angle, or two side lengths, calculates the rest of the side lengths
// and angles of a right triangle. Returns [ADJACENT, OPPOSITE, HYPOTENUSE, ANGLE, ANGLE2] where // and angles of a right triangle. Returns [ADJACENT, OPPOSITE, HYPOTENUSE, ANGLE, ANGLE2] where
@ -628,6 +649,7 @@ function tri_calc(ang,ang2,adj,opp,hyp) =
// Alias: opp_hyp_to_adj() // Alias: opp_hyp_to_adj()
// Usage: // Usage:
// adj = hyp_opp_to_adj(hyp,opp); // adj = hyp_opp_to_adj(hyp,opp);
// Topics: Geometry, Triangles
// Description: // Description:
// Given the lengths of the hypotenuse and opposite side of a right triangle, returns the length // Given the lengths of the hypotenuse and opposite side of a right triangle, returns the length
// of the adjacent side. // of the adjacent side.
@ -648,6 +670,7 @@ function opp_hyp_to_adj(opp,hyp) = hyp_opp_to_adj(hyp,opp);
// Alias: ang_hyp_to_adj() // Alias: ang_hyp_to_adj()
// Usage: // Usage:
// adj = hyp_ang_to_adj(hyp,ang); // adj = hyp_ang_to_adj(hyp,ang);
// Topics: Geometry, Triangles
// Description: // Description:
// Given the length of the hypotenuse and the angle of the primary corner of a right triangle, // Given the length of the hypotenuse and the angle of the primary corner of a right triangle,
// returns the length of the adjacent side. // returns the length of the adjacent side.
@ -668,6 +691,7 @@ function ang_hyp_to_adj(ang,hyp) = hyp_ang_to_adj(hyp, ang);
// Alias: ang_opp_to_adj() // Alias: ang_opp_to_adj()
// Usage: // Usage:
// adj = opp_ang_to_adj(opp,ang); // adj = opp_ang_to_adj(opp,ang);
// Topics: Geometry, Triangles
// Description: // Description:
// Given the angle of the primary corner of a right triangle, and the length of the side opposite of it, // Given the angle of the primary corner of a right triangle, and the length of the side opposite of it,
// returns the length of the adjacent side. // returns the length of the adjacent side.
@ -688,6 +712,7 @@ function ang_opp_to_adj(ang,opp) = opp_ang_to_adj(opp,ang);
// Alias: adj_hyp_to_opp() // Alias: adj_hyp_to_opp()
// Usage: // Usage:
// opp = hyp_adj_to_opp(hyp,adj); // opp = hyp_adj_to_opp(hyp,adj);
// Topics: Geometry, Triangles
// Description: // Description:
// Given the length of the hypotenuse and the adjacent side, returns the length of the opposite side. // Given the length of the hypotenuse and the adjacent side, returns the length of the opposite side.
// Arguments: // Arguments:
@ -707,6 +732,7 @@ function adj_hyp_to_opp(adj,hyp) = hyp_adj_to_opp(hyp,adj);
// Alias: ang_hyp_to_opp() // Alias: ang_hyp_to_opp()
// Usage: // Usage:
// opp = hyp_ang_to_opp(hyp,adj); // opp = hyp_ang_to_opp(hyp,adj);
// Topics: Geometry, Triangles
// Description: // Description:
// Given the length of the hypotenuse of a right triangle, and the angle of the corner, returns the length of the opposite side. // Given the length of the hypotenuse of a right triangle, and the angle of the corner, returns the length of the opposite side.
// Arguments: // Arguments:
@ -726,6 +752,7 @@ function ang_hyp_to_opp(ang,hyp) = hyp_ang_to_opp(hyp,ang);
// Alias: ang_adj_to_opp() // Alias: ang_adj_to_opp()
// Usage: // Usage:
// opp = adj_ang_to_opp(adj,ang); // opp = adj_ang_to_opp(adj,ang);
// Topics: Geometry, Triangles
// Description: // Description:
// Given the length of the adjacent side of a right triangle, and the angle of the corner, returns the length of the opposite side. // Given the length of the adjacent side of a right triangle, and the angle of the corner, returns the length of the opposite side.
// Arguments: // Arguments:
@ -745,6 +772,7 @@ function ang_adj_to_opp(ang,adj) = adj_ang_to_opp(adj,ang);
// Alias: opp_adj_to_hyp() // Alias: opp_adj_to_hyp()
// Usage: // Usage:
// hyp = adj_opp_to_hyp(adj,opp); // hyp = adj_opp_to_hyp(adj,opp);
// Topics: Geometry, Triangles
// Description: // Description:
// Given the length of the adjacent and opposite sides of a right triangle, returns the length of thee hypotenuse. // Given the length of the adjacent and opposite sides of a right triangle, returns the length of thee hypotenuse.
// Arguments: // Arguments:
@ -764,6 +792,7 @@ function opp_adj_to_hyp(opp,adj) = adj_opp_to_hyp(adj,opp);
// Alias: ang_adj_to_hyp() // Alias: ang_adj_to_hyp()
// Usage: // Usage:
// hyp = adj_ang_to_hyp(adj,ang); // hyp = adj_ang_to_hyp(adj,ang);
// Topics: Geometry, Triangles
// Description: // Description:
// For a right triangle, given the length of the adjacent side, and the corner angle, returns the length of the hypotenuse. // For a right triangle, given the length of the adjacent side, and the corner angle, returns the length of the hypotenuse.
// Arguments: // Arguments:
@ -783,6 +812,7 @@ function ang_adj_to_hyp(ang,adj) = adj_ang_to_hyp(adj,ang);
// Alias: ang_opp_to_hyp() // Alias: ang_opp_to_hyp()
// Usage: // Usage:
// hyp = opp_ang_to_hyp(opp,ang); // hyp = opp_ang_to_hyp(opp,ang);
// Topics: Geometry, Triangles
// Description: // Description:
// For a right triangle, given the length of the opposite side, and the corner angle, returns the length of the hypotenuse. // For a right triangle, given the length of the opposite side, and the corner angle, returns the length of the hypotenuse.
// Arguments: // Arguments:
@ -821,6 +851,7 @@ function adj_hyp_to_ang(adj,hyp) = hyp_adj_to_ang(hyp,adj);
// Alias: opp_hyp_to_ang() // Alias: opp_hyp_to_ang()
// Usage: // Usage:
// ang = hyp_opp_to_ang(hyp,opp); // ang = hyp_opp_to_ang(hyp,opp);
// Topics: Geometry, Triangles
// Description: // Description:
// For a right triangle, given the lengths of the hypotenuse and the opposite sides, returns the angle of the corner. // For a right triangle, given the lengths of the hypotenuse and the opposite sides, returns the angle of the corner.
// Arguments: // Arguments:
@ -840,6 +871,7 @@ function opp_hyp_to_ang(opp,hyp) = hyp_opp_to_ang(hyp,opp);
// Alias: opp_adj_to_ang() // Alias: opp_adj_to_ang()
// Usage: // Usage:
// ang = adj_opp_to_ang(adj,opp); // ang = adj_opp_to_ang(adj,opp);
// Topics: Geometry, Triangles
// Description: // Description:
// For a right triangle, given the lengths of the adjacent and opposite sides, returns the angle of the corner. // For a right triangle, given the lengths of the adjacent and opposite sides, returns the angle of the corner.
// Arguments: // Arguments:
@ -857,7 +889,8 @@ function opp_adj_to_ang(opp,adj) = adj_opp_to_ang(adj,opp);
// Function: triangle_area() // Function: triangle_area()
// Usage: // Usage:
// triangle_area(a,b,c); // area = triangle_area(a,b,c);
// Topics: Geometry, Triangles, Area
// Description: // Description:
// Returns the area of a triangle formed between three 2D or 3D vertices. // Returns the area of a triangle formed between three 2D or 3D vertices.
// Result will be negative if the points are 2D and in clockwise order. // Result will be negative if the points are 2D and in clockwise order.
@ -881,7 +914,8 @@ function triangle_area(a,b,c) =
// Function: plane3pt() // Function: plane3pt()
// Usage: // Usage:
// plane3pt(p1, p2, p3); // plane = plane3pt(p1, p2, p3);
// Topics: Geometry, Planes
// Description: // Description:
// Generates the normalized cartesian equation of a plane from three 3d points. // Generates the normalized cartesian equation of a plane from three 3d points.
// Returns [A,B,C,D] where Ax + By + Cz = D is the equation of a plane. // Returns [A,B,C,D] where Ax + By + Cz = D is the equation of a plane.
@ -902,7 +936,8 @@ function plane3pt(p1, p2, p3) =
// Function: plane3pt_indexed() // Function: plane3pt_indexed()
// Usage: // Usage:
// plane3pt_indexed(points, i1, i2, i3); // plane = plane3pt_indexed(points, i1, i2, i3);
// Topics: Geometry, Planes
// Description: // Description:
// Given a list of 3d points, and the indices of three of those points, // Given a list of 3d points, and the indices of three of those points,
// generates the normalized cartesian equation of a plane that those points all // generates the normalized cartesian equation of a plane that those points all
@ -927,7 +962,8 @@ function plane3pt_indexed(points, i1, i2, i3) =
// Function: plane_from_normal() // Function: plane_from_normal()
// Usage: // Usage:
// plane_from_normal(normal, [pt]) // plane = plane_from_normal(normal, [pt])
// Topics: Geometry, Planes
// Description: // Description:
// Returns a plane defined by a normal vector and a point. // Returns a plane defined by a normal vector and a point.
// Arguments: // Arguments:
@ -985,7 +1021,8 @@ function _covariance_evec_eval(points) =
// Function: plane_from_points() // Function: plane_from_points()
// Usage: // Usage:
// plane_from_points(points, <fast>, <eps>); // plane = plane_from_points(points, <fast>, <eps>);
// Topics: Geometry, Planes, Points
// Description: // Description:
// Given a list of 3 or more coplanar 3D points, returns the coefficients of the normalized cartesian equation of a plane, // Given a list of 3 or more coplanar 3D points, returns the coefficients of the normalized cartesian equation of a plane,
// that is [A,B,C,D] where Ax+By+Cz=D is the equation of the plane and norm([A,B,C])=1. // that is [A,B,C,D] where Ax+By+Cz=D is the equation of the plane and norm([A,B,C])=1.
@ -1020,7 +1057,8 @@ function plane_from_points(points, fast=false, eps=EPSILON) =
// Function: plane_from_polygon() // Function: plane_from_polygon()
// Usage: // Usage:
// plane_from_polygon(points, [fast], [eps]); // plane = plane_from_polygon(points, [fast], [eps]);
// Topics: Geometry, Planes, Polygons
// Description: // Description:
// Given a 3D planar polygon, returns the normalized cartesian equation of its plane. // Given a 3D planar polygon, returns the normalized cartesian equation of its plane.
// Returns [A,B,C,D] where Ax+By+Cz=D is the equation of the plane where norm([A,B,C])=1. // Returns [A,B,C,D] where Ax+By+Cz=D is the equation of the plane where norm([A,B,C])=1.
@ -1049,7 +1087,8 @@ function plane_from_polygon(poly, fast=false, eps=EPSILON) =
// Function: plane_normal() // Function: plane_normal()
// Usage: // Usage:
// plane_normal(plane); // vec = plane_normal(plane);
// Topics: Geometry, Planes
// Description: // Description:
// Returns the unit length normal vector for the given plane. // Returns the unit length normal vector for the given plane.
// Arguments: // Arguments:
@ -1062,6 +1101,7 @@ function plane_normal(plane) =
// Function: plane_offset() // Function: plane_offset()
// Usage: // Usage:
// d = plane_offset(plane); // d = plane_offset(plane);
// Topics: Geometry, Planes
// Description: // Description:
// Returns coeficient D of the normalized plane equation `Ax+By+Cz=D`, or the scalar offset of the plane from the origin. // Returns coeficient D of the normalized plane equation `Ax+By+Cz=D`, or the scalar offset of the plane from the origin.
// This value may be negative. // This value may be negative.
@ -1077,6 +1117,7 @@ function plane_offset(plane) =
// Function: projection_on_plane() // Function: projection_on_plane()
// Usage: // Usage:
// pts = projection_on_plane(plane, points); // pts = projection_on_plane(plane, points);
// Topics: Geometry, Planes, Projection
// Description: // Description:
// Given a plane definition `[A,B,C,D]`, where `Ax+By+Cz=D`, and a list of 2d or // Given a plane definition `[A,B,C,D]`, where `Ax+By+Cz=D`, and a list of 2d or
// 3d points, return the 3D orthogonal projection of the points on the plane. // 3d points, return the 3D orthogonal projection of the points on the plane.
@ -1112,6 +1153,7 @@ function projection_on_plane(plane, points) =
// Function: plane_point_nearest_origin() // Function: plane_point_nearest_origin()
// Usage: // Usage:
// pt = plane_point_nearest_origin(plane); // pt = plane_point_nearest_origin(plane);
// Topics: Geometry, Planes, Distance
// Description: // Description:
// Returns the point on the plane that is closest to the origin. // Returns the point on the plane that is closest to the origin.
// Arguments: // Arguments:
@ -1123,7 +1165,8 @@ function plane_point_nearest_origin(plane) =
// Function: point_plane_distance() // Function: point_plane_distance()
// Usage: // Usage:
// point_plane_distance(plane, point) // dist = point_plane_distance(plane, point)
// Topics: Geometry, Planes, Distance
// Description: // Description:
// Given a plane as [A,B,C,D] where the cartesian equation for that plane // Given a plane as [A,B,C,D] where the cartesian equation for that plane
// is Ax+By+Cz=D, determines how far from that plane the given point is. // is Ax+By+Cz=D, determines how far from that plane the given point is.
@ -1159,6 +1202,7 @@ function _general_plane_line_intersection(plane, line, eps=EPSILON) =
// Function: normalize_plane() // Function: normalize_plane()
// Usage: // Usage:
// nplane = normalize_plane(plane); // nplane = normalize_plane(plane);
// Topics: Geometry, Planes
// Description: // Description:
// Returns a new representation [A,B,C,D] of `plane` where norm([A,B,C]) is equal to one. // Returns a new representation [A,B,C,D] of `plane` where norm([A,B,C]) is equal to one.
function normalize_plane(plane) = function normalize_plane(plane) =
@ -1169,6 +1213,7 @@ function normalize_plane(plane) =
// Function: plane_line_angle() // Function: plane_line_angle()
// Usage: // Usage:
// angle = plane_line_angle(plane,line); // angle = plane_line_angle(plane,line);
// Topics: Geometry, Planes, Lines, Angle
// Description: // Description:
// Compute the angle between a plane [A, B, C, D] and a 3d line, specified as a pair of 3d points [p1,p2]. // Compute the angle between a plane [A, B, C, D] and a 3d line, specified as a pair of 3d points [p1,p2].
// The resulting angle is signed, with the sign positive if the vector p2-p1 lies on // The resulting angle is signed, with the sign positive if the vector p2-p1 lies on
@ -1187,6 +1232,7 @@ function plane_line_angle(plane, line) =
// Function: plane_line_intersection() // Function: plane_line_intersection()
// Usage: // Usage:
// pt = plane_line_intersection(plane, line, [bounded], [eps]); // pt = plane_line_intersection(plane, line, [bounded], [eps]);
// Topics: Geometry, Planes, Lines, Intersection
// Description: // Description:
// Takes a line, and a plane [A,B,C,D] where the equation of that plane is `Ax+By+Cz=D`. // Takes a line, and a plane [A,B,C,D] where the equation of that plane is `Ax+By+Cz=D`.
// If `line` intersects `plane` at one point, then that intersection point is returned. // If `line` intersects `plane` at one point, then that intersection point is returned.
@ -1214,6 +1260,7 @@ function plane_line_intersection(plane, line, bounded=false, eps=EPSILON) =
// Function: polygon_line_intersection() // Function: polygon_line_intersection()
// Usage: // Usage:
// pt = polygon_line_intersection(poly, line, [bounded], [eps]); // pt = polygon_line_intersection(poly, line, [bounded], [eps]);
// Topics: Geometry, Polygons, Lines, Intersection
// Description: // Description:
// Takes a possibly bounded line, and a 3D planar polygon, and finds their intersection point. // Takes a possibly bounded line, and a 3D planar polygon, and finds their intersection point.
// If the line and the polygon are on the same plane then returns a list, possibly empty, of 3D line // If the line and the polygon are on the same plane then returns a list, possibly empty, of 3D line
@ -1269,7 +1316,9 @@ function polygon_line_intersection(poly, line, bounded=false, eps=EPSILON) =
// Function: plane_intersection() // Function: plane_intersection()
// Usage: // Usage:
// plane_intersection(plane1, plane2, [plane3]) // line = plane_intersection(plane1, plane2)
// pt = plane_intersection(plane1, plane2, plane3)
// Topics: Geometry, Planes, Intersection
// Description: // Description:
// Compute the point which is the intersection of the three planes, or the line intersection of two planes. // Compute the point which is the intersection of the three planes, or the line intersection of two planes.
// If you give three planes the intersection is returned as a point. If you give two planes the intersection // If you give three planes the intersection is returned as a point. If you give two planes the intersection
@ -1301,7 +1350,8 @@ function plane_intersection(plane1,plane2,plane3) =
// Function: coplanar() // Function: coplanar()
// Usage: // Usage:
// coplanar(points,<eps>); // test = coplanar(points,<eps>);
// Topics: Geometry, Coplanarity
// Description: // Description:
// Returns true if the given 3D points are non-collinear and are on a plane. // Returns true if the given 3D points are non-collinear and are on a plane.
// Arguments: // Arguments:
@ -1328,7 +1378,8 @@ function _pointlist_greatest_distance(points,plane) =
// Function: points_on_plane() // Function: points_on_plane()
// Usage: // Usage:
// points_on_plane(points, plane, <eps>); // test = points_on_plane(points, plane, <eps>);
// Topics: Geometry, Planes, Points
// Description: // Description:
// Returns true if the given 3D points are on the given plane. // Returns true if the given 3D points are on the given plane.
// Arguments: // Arguments:
@ -1344,7 +1395,8 @@ function points_on_plane(points, plane, eps=EPSILON) =
// Function: in_front_of_plane() // Function: in_front_of_plane()
// Usage: // Usage:
// in_front_of_plane(plane, point); // test = in_front_of_plane(plane, point);
// Topics: Geometry, Planes
// Description: // Description:
// Given a plane as [A,B,C,D] where the cartesian equation for that plane // Given a plane as [A,B,C,D] where the cartesian equation for that plane
// is Ax+By+Cz=D, determines if the given 3D point is on the side of that // is Ax+By+Cz=D, determines if the given 3D point is on the side of that
@ -1363,6 +1415,7 @@ function in_front_of_plane(plane, point) =
// Function&Module: circle_2tangents() // Function&Module: circle_2tangents()
// Usage: As Function // Usage: As Function
// circ = circle_2tangents(pt1, pt2, pt3, r|d, <tangents>); // circ = circle_2tangents(pt1, pt2, pt3, r|d, <tangents>);
// Topics: Geometry, Circles, Tangents
// Usage: As Module // Usage: As Module
// circle_2tangents(pt1, pt2, pt3, r|d, <h>, <center>); // circle_2tangents(pt1, pt2, pt3, r|d, <h>, <center>);
// Description: // Description:
@ -1475,6 +1528,7 @@ module circle_2tangents(pt1, pt2, pt3, r, d, h, center=false) {
// Usage: As Function // Usage: As Function
// circ = circle_3points(pt1, pt2, pt3); // circ = circle_3points(pt1, pt2, pt3);
// circ = circle_3points([pt1, pt2, pt3]); // circ = circle_3points([pt1, pt2, pt3]);
// Topics: Geometry, Circles
// Usage: As Module // Usage: As Module
// circle_3points(pt1, pt2, pt3, <h>, <center>); // circle_3points(pt1, pt2, pt3, <h>, <center>);
// circle_3points([pt1, pt2, pt3], <h>, <center>); // circle_3points([pt1, pt2, pt3], <h>, <center>);
@ -1553,6 +1607,7 @@ module circle_3points(pt1, pt2, pt3, h, center=false) {
// Function: circle_point_tangents() // Function: circle_point_tangents()
// Usage: // Usage:
// tangents = circle_point_tangents(r|d, cp, pt); // tangents = circle_point_tangents(r|d, cp, pt);
// Topics: Geometry, Circles, Tangents
// Description: // Description:
// Given a 2d circle and a 2d point outside that circle, finds the 2d tangent point(s) on the circle for a // Given a 2d circle and a 2d point outside that circle, finds the 2d tangent point(s) on the circle for a
// line passing through the point. Returns a list of zero or more 2D tangent points. // line passing through the point. Returns a list of zero or more 2D tangent points.
@ -1587,6 +1642,7 @@ function circle_point_tangents(r, d, cp, pt) =
// Function: circle_circle_tangents() // Function: circle_circle_tangents()
// Usage: // Usage:
// segs = circle_circle_tangents(c1, r1|d1, c2, r2|d2); // segs = circle_circle_tangents(c1, r1|d1, c2, r2|d2);
// Topics: Geometry, Circles, Tangents
// Description: // Description:
// Computes 2d lines tangents to a pair of circles in 2d. Returns a list of line endpoints [p1,p2] where // Computes 2d lines tangents to a pair of circles in 2d. Returns a list of line endpoints [p1,p2] where
// p2 is the tangent point on circle 1 and p2 is the tangent point on circle 2. // p2 is the tangent point on circle 1 and p2 is the tangent point on circle 2.
@ -1669,6 +1725,7 @@ function circle_circle_tangents(c1,r1,c2,r2,d1,d2) =
// Function: circle_line_intersection() // Function: circle_line_intersection()
// Usage: // Usage:
// isect = circle_line_intersection(c,<r|d>,<line>,<bounded>,<eps>); // isect = circle_line_intersection(c,<r|d>,<line>,<bounded>,<eps>);
// Topics: Geometry, Circles, Lines, Intersection
// Description: // Description:
// Find intersection points between a 2d circle and a line, ray or segment specified by two points. // Find intersection points between a 2d circle and a line, ray or segment specified by two points.
// By default the line is unbounded. // By default the line is unbounded.
@ -1709,7 +1766,8 @@ function circle_line_intersection(c,r,d,line,bounded=false,eps=EPSILON) =
// Function: noncollinear_triple() // Function: noncollinear_triple()
// Usage: // Usage:
// noncollinear_triple(points); // test = noncollinear_triple(points);
// Topics: Geometry, Noncollinearity
// Description: // Description:
// Finds the indices of three good non-collinear points from the pointlist `points`. // Finds the indices of three good non-collinear points from the pointlist `points`.
// If all points are collinear returns [] when `error=true` or an error otherwise . // If all points are collinear returns [] when `error=true` or an error otherwise .
@ -1739,7 +1797,8 @@ function noncollinear_triple(points,error=true,eps=EPSILON) =
// Function: pointlist_bounds() // Function: pointlist_bounds()
// Usage: // Usage:
// pointlist_bounds(pts); // pt_pair = pointlist_bounds(pts);
// Topics: Geometry, Bounding Boxes, Bounds
// Description: // Description:
// Finds the bounds containing all the points in `pts` which can be a list of points in any dimension. // Finds the bounds containing all the points in `pts` which can be a list of points in any dimension.
// Returns a list of two items: a list of the minimums and a list of the maximums. For example, with // Returns a list of two items: a list of the minimums and a list of the maximums. For example, with
@ -1760,7 +1819,8 @@ function pointlist_bounds(pts) =
// Function: closest_point() // Function: closest_point()
// Usage: // Usage:
// closest_point(pt, points); // index = closest_point(pt, points);
// Topics: Geometry, Points, Distance
// Description: // Description:
// Given a list of `points`, finds the index of the closest point to `pt`. // Given a list of `points`, finds the index of the closest point to `pt`.
// Arguments: // Arguments:
@ -1774,7 +1834,8 @@ function closest_point(pt, points) =
// Function: furthest_point() // Function: furthest_point()
// Usage: // Usage:
// furthest_point(pt, points); // index = furthest_point(pt, points);
// Topics: Geometry, Points, Distance
// Description: // Description:
// Given a list of `points`, finds the index of the furthest point from `pt`. // Given a list of `points`, finds the index of the furthest point from `pt`.
// Arguments: // Arguments:
@ -1792,6 +1853,7 @@ function furthest_point(pt, points) =
// Function: polygon_area() // Function: polygon_area()
// Usage: // Usage:
// area = polygon_area(poly); // area = polygon_area(poly);
// Topics: Geometry, Polygons, Area
// Description: // Description:
// Given a 2D or 3D planar polygon, returns the area of that polygon. // Given a 2D or 3D planar polygon, returns the area of that polygon.
// If the polygon is self-crossing, the results are undefined. For non-planar 3D polygon the result is `undef`. // If the polygon is self-crossing, the results are undefined. For non-planar 3D polygon the result is `undef`.
@ -1808,18 +1870,19 @@ function polygon_area(poly, signed=false) =
: let( plane = plane_from_polygon(poly) ) : let( plane = plane_from_polygon(poly) )
plane==[]? undef : plane==[]? undef :
let( let(
n = plane_normal(plane), n = plane_normal(plane),
total = sum([ total =
for(i=[1:1:len(poly)-2]) sum([ for(i=[1:1:len(poly)-2])
cross(poly[i]-poly[0], poly[i+1]-poly[0]) cross(poly[i]-poly[0], poly[i+1]-poly[0])
]) * n/2 ]) * n/2
) )
signed ? total : abs(total); signed ? total : abs(total);
// Function: polygon_shift() // Function: polygon_shift()
// Usage: // Usage:
// polygon_shift(poly, i); // newpoly = polygon_shift(poly, i);
// Topics: Geometry, Polygons
// Description: // Description:
// Given a polygon `poly`, rotates the point ordering so that the first point in the polygon path is the one at index `i`. // Given a polygon `poly`, rotates the point ordering so that the first point in the polygon path is the one at index `i`.
// Arguments: // Arguments:
@ -1834,7 +1897,8 @@ function polygon_shift(poly, i) =
// Function: polygon_shift_to_closest_point() // Function: polygon_shift_to_closest_point()
// Usage: // Usage:
// polygon_shift_to_closest_point(path, pt); // newpoly = polygon_shift_to_closest_point(path, pt);
// Topics: Geometry, Polygons
// Description: // Description:
// Given a polygon `poly`, rotates the point ordering so that the first point in the path is the one closest to the given point `pt`. // Given a polygon `poly`, rotates the point ordering so that the first point in the path is the one closest to the given point `pt`.
// Arguments: // Arguments:
@ -1853,6 +1917,7 @@ function polygon_shift_to_closest_point(poly, pt) =
// Function: reindex_polygon() // Function: reindex_polygon()
// Usage: // Usage:
// newpoly = reindex_polygon(reference, poly); // newpoly = reindex_polygon(reference, poly);
// Topics: Geometry, Polygons
// Description: // Description:
// Rotates and possibly reverses the point order of a 2d or 3d polygon path to optimize its pairwise point // Rotates and possibly reverses the point order of a 2d or 3d polygon path to optimize its pairwise point
// association with a reference polygon. The two polygons must have the same number of vertices and be the same dimension. // association with a reference polygon. The two polygons must have the same number of vertices and be the same dimension.
@ -1903,6 +1968,7 @@ function reindex_polygon(reference, poly, return_error=false) =
// Function: align_polygon() // Function: align_polygon()
// Usage: // Usage:
// newpoly = align_polygon(reference, poly, angles, <cp>); // newpoly = align_polygon(reference, poly, angles, <cp>);
// Topics: Geometry, Polygons
// Description: // Description:
// Tries the list or range of angles to find a rotation of the specified 2D polygon that best aligns // Tries the list or range of angles to find a rotation of the specified 2D polygon that best aligns
// with the reference 2D polygon. For each angle, the polygon is reindexed, which is a costly operation // with the reference 2D polygon. For each angle, the polygon is reindexed, which is a costly operation
@ -1940,7 +2006,8 @@ function align_polygon(reference, poly, angles, cp) =
// Function: centroid() // Function: centroid()
// Usage: // Usage:
// cp = centroid(poly); // cpt = centroid(poly);
// Topics: Geometry, Polygons, Centroid
// Description: // Description:
// Given a simple 2D polygon, returns the 2D coordinates of the polygon's centroid. // Given a simple 2D polygon, returns the 2D coordinates of the polygon's centroid.
// Given a simple 3D planar polygon, returns the 3D coordinates of the polygon's centroid. // Given a simple 3D planar polygon, returns the 3D coordinates of the polygon's centroid.
@ -1974,7 +2041,8 @@ function centroid(poly, eps=EPSILON) =
// Function: point_in_polygon() // Function: point_in_polygon()
// Usage: // Usage:
// point_in_polygon(point, poly, <eps>) // test = point_in_polygon(point, poly, <eps>)
// Topics: Geometry, Polygons
// Description: // Description:
// This function tests whether the given 2D point is inside, outside or on the boundary of // This function tests whether the given 2D point is inside, outside or on the boundary of
// the specified 2D polygon using either the Nonzero Winding rule or the Even-Odd rule. // the specified 2D polygon using either the Nonzero Winding rule or the Even-Odd rule.
@ -2035,7 +2103,8 @@ function point_in_polygon(point, poly, nonzero=true, eps=EPSILON) =
// Function: polygon_is_clockwise() // Function: polygon_is_clockwise()
// Usage: // Usage:
// polygon_is_clockwise(poly); // test = polygon_is_clockwise(poly);
// Topics: Geometry, Polygons, Clockwise
// Description: // Description:
// Return true if the given 2D simple polygon is in clockwise order, false otherwise. // Return true if the given 2D simple polygon is in clockwise order, false otherwise.
// Results for complex (self-intersecting) polygon are indeterminate. // Results for complex (self-intersecting) polygon are indeterminate.
@ -2048,7 +2117,8 @@ function polygon_is_clockwise(poly) =
// Function: clockwise_polygon() // Function: clockwise_polygon()
// Usage: // Usage:
// clockwise_polygon(poly); // newpoly = clockwise_polygon(poly);
// Topics: Geometry, Polygons, Clockwise
// Description: // Description:
// Given a 2D polygon path, returns the clockwise winding version of that path. // Given a 2D polygon path, returns the clockwise winding version of that path.
// Arguments: // Arguments:
@ -2060,7 +2130,8 @@ function clockwise_polygon(poly) =
// Function: ccw_polygon() // Function: ccw_polygon()
// Usage: // Usage:
// ccw_polygon(poly); // newpoly = ccw_polygon(poly);
// Topics: Geometry, Polygons, Clockwise
// Description: // Description:
// Given a 2D polygon poly, returns the counter-clockwise winding version of that poly. // Given a 2D polygon poly, returns the counter-clockwise winding version of that poly.
// Arguments: // Arguments:
@ -2072,7 +2143,8 @@ function ccw_polygon(poly) =
// Function: reverse_polygon() // Function: reverse_polygon()
// Usage: // Usage:
// reverse_polygon(poly) // newpoly = reverse_polygon(poly)
// Topics: Geometry, Polygons, Clockwise
// Description: // Description:
// Reverses a polygon's winding direction, while still using the same start point. // Reverses a polygon's winding direction, while still using the same start point.
// Arguments: // Arguments:
@ -2084,7 +2156,8 @@ function reverse_polygon(poly) =
// Function: polygon_normal() // Function: polygon_normal()
// Usage: // Usage:
// n = polygon_normal(poly); // vec = polygon_normal(poly);
// Topics: Geometry, Polygons
// Description: // Description:
// Given a 3D planar polygon, returns a unit-length normal vector for the // Given a 3D planar polygon, returns a unit-length normal vector for the
// clockwise orientation of the polygon. If the polygon points are collinear, returns `undef`. // clockwise orientation of the polygon. If the polygon points are collinear, returns `undef`.
@ -2192,6 +2265,7 @@ function _split_polygon_at_z(poly, z) =
// Function: split_polygons_at_each_x() // Function: split_polygons_at_each_x()
// Usage: // Usage:
// splitpolys = split_polygons_at_each_x(polys, xs); // splitpolys = split_polygons_at_each_x(polys, xs);
// Topics: Geometry, Polygons, Intersections
// Description: // Description:
// Given a list of 3D polygons, splits all of them wherever they cross any X value given in `xs`. // Given a list of 3D polygons, splits all of them wherever they cross any X value given in `xs`.
// Arguments: // Arguments:
@ -2212,6 +2286,7 @@ function split_polygons_at_each_x(polys, xs, _i=0) =
// Function: split_polygons_at_each_y() // Function: split_polygons_at_each_y()
// Usage: // Usage:
// splitpolys = split_polygons_at_each_y(polys, ys); // splitpolys = split_polygons_at_each_y(polys, ys);
// Topics: Geometry, Polygons, Intersections
// Description: // Description:
// Given a list of 3D polygons, splits all of them wherever they cross any Y value given in `ys`. // Given a list of 3D polygons, splits all of them wherever they cross any Y value given in `ys`.
// Arguments: // Arguments:
@ -2232,6 +2307,7 @@ function split_polygons_at_each_y(polys, ys, _i=0) =
// Function: split_polygons_at_each_z() // Function: split_polygons_at_each_z()
// Usage: // Usage:
// splitpolys = split_polygons_at_each_z(polys, zs); // splitpolys = split_polygons_at_each_z(polys, zs);
// Topics: Geometry, Polygons, Intersections
// Description: // Description:
// Given a list of 3D polygons, splits all of them wherever they cross any Z value given in `zs`. // Given a list of 3D polygons, splits all of them wherever they cross any Z value given in `zs`.
// Arguments: // Arguments:
@ -2255,7 +2331,8 @@ function split_polygons_at_each_z(polys, zs, _i=0) =
// Function: is_convex_polygon() // Function: is_convex_polygon()
// Usage: // Usage:
// is_convex_polygon(poly); // test = is_convex_polygon(poly);
// Topics: Geometry, Convexity, Test
// Description: // Description:
// Returns true if the given 2D or 3D polygon is convex. // Returns true if the given 2D or 3D polygon is convex.
// The result is meaningless if the polygon is not simple (self-intersecting) or non coplanar. // The result is meaningless if the polygon is not simple (self-intersecting) or non coplanar.
@ -2264,11 +2341,11 @@ function split_polygons_at_each_z(polys, zs, _i=0) =
// poly = Polygon to check. // poly = Polygon to check.
// eps = Tolerance for the collinearity test. Default: EPSILON. // eps = Tolerance for the collinearity test. Default: EPSILON.
// Example: // Example:
// is_convex_polygon(circle(d=50)); // Returns: true // test1 = is_convex_polygon(circle(d=50)); // Returns: true
// is_convex_polygon(rot([50,120,30], p=path3d(circle(1,$fn=50)))); // Returns: true // test2 = is_convex_polygon(rot([50,120,30], p=path3d(circle(1,$fn=50)))); // Returns: true
// Example: // Example:
// spiral = [for (i=[0:36]) let(a=-i*10) (10+i)*[cos(a),sin(a)]]; // spiral = [for (i=[0:36]) let(a=-i*10) (10+i)*[cos(a),sin(a)]];
// is_convex_polygon(spiral); // Returns: false // test = is_convex_polygon(spiral); // Returns: false
function is_convex_polygon(poly,eps=EPSILON) = function is_convex_polygon(poly,eps=EPSILON) =
assert(is_path(poly), "The input should be a 2D or 3D polygon." ) assert(is_path(poly), "The input should be a 2D or 3D polygon." )
let( let(
@ -2289,9 +2366,10 @@ function is_convex_polygon(poly,eps=EPSILON) =
// Function: convex_distance() // Function: convex_distance()
// Usage: // Usage:
// convex_distance(pts1, pts2,<eps=>); // dist = convex_distance(pts1, pts2,<eps=>);
// See also: // Topics: Geometry, Convexity, Distance
// convex_collision // See also:
// convex_collision(), hull()
// Description: // Description:
// Returns the smallest distance between a point in convex hull of `points1` // Returns the smallest distance between a point in convex hull of `points1`
// and a point in the convex hull of `points2`. All the points in the lists // and a point in the convex hull of `points2`. All the points in the lists
@ -2349,9 +2427,10 @@ function _GJK_distance(points1, points2, eps=EPSILON, lbd, d, simplex=[]) =
// Function: convex_collision() // Function: convex_collision()
// Usage: // Usage:
// convex_collision(pts1, pts2,<eps=>); // test = convex_collision(pts1, pts2,<eps=>);
// See also: // Topics: Geometry, Convexity, Collision, Intersection
// convex_distance // See also:
// convex_distance(), hull()
// Description: // Description:
// Returns `true` if the convex hull of `points1` intercepts the convex hull of `points2` // Returns `true` if the convex hull of `points1` intercepts the convex hull of `points2`
// otherwise, `false`. // otherwise, `false`.
@ -2412,7 +2491,7 @@ function _closest_simplex(s,eps=EPSILON) =
: _closest_s3(s,eps); : _closest_s3(s,eps);
// find the closest to a 1-simplex // find the point of a 1-simplex closest to the origin
// Based on: http://uu.diva-portal.org/smash/get/diva2/FFULLTEXT01.pdf // Based on: http://uu.diva-portal.org/smash/get/diva2/FFULLTEXT01.pdf
function _closest_s1(s,eps=EPSILON) = function _closest_s1(s,eps=EPSILON) =
norm(s[1]-s[0])<eps*(norm(s[0])+norm(s[1]))/2 ? [ s[0], [s[0]] ] : norm(s[1]-s[0])<eps*(norm(s[0])+norm(s[1]))/2 ? [ s[0], [s[0]] ] :
@ -2425,7 +2504,7 @@ function _closest_s1(s,eps=EPSILON) =
[ s[0]+t*c, s ]; [ s[0]+t*c, s ];
// find the closest to a 2-simplex // find the point of a 2-simplex closest to the origin
// Based on: http://uu.diva-portal.org/smash/get/diva2/FFULLTEXT01.pdf // Based on: http://uu.diva-portal.org/smash/get/diva2/FFULLTEXT01.pdf
function _closest_s2(s,eps=EPSILON) = function _closest_s2(s,eps=EPSILON) =
let( let(
@ -2461,7 +2540,7 @@ function _closest_s2(s,eps=EPSILON) =
c*(c-a)>0 ? _closest_s1([s[0],s[2]],eps) : _closest_s1([s[1],s[2]],eps); c*(c-a)>0 ? _closest_s1([s[0],s[2]],eps) : _closest_s1([s[1],s[2]],eps);
// find the closest to a 3-simplex // find the point of a 3-simplex closest to the origin
// it seems that degenerate 3-simplices are correctly manage without extra code // it seems that degenerate 3-simplices are correctly manage without extra code
function _closest_s3(s,eps=EPSILON) = function _closest_s3(s,eps=EPSILON) =
assert( len(s[0])==3 && len(s)==4, "Internal error." ) assert( len(s[0])==3 && len(s)==4, "Internal error." )
@ -2488,7 +2567,7 @@ function _closest_s3(s,eps=EPSILON) =
let( // look for the origin-facing tri closest to the origin let( // look for the origin-facing tri closest to the origin
closest = [for(i=facing) _closest_s2(tris[i], eps) ], closest = [for(i=facing) _closest_s2(tris[i], eps) ],
dist = [for(cl=closest) norm(cl[0]) ], dist = [for(cl=closest) norm(cl[0]) ],
nearest = min_index(dist) nearest = min_index(dist)
) )
closest[nearest]; closest[nearest];