BOSL2/geometry.scad

//////////////////////////////////////////////////////////////////////
// LibFile: geometry.scad
//   Geometry helpers.
//   To use, add the following lines to the beginning of your file:
//   ```
//   use <BOSL2/std.scad>
//   ```
//////////////////////////////////////////////////////////////////////

/*
BSD 2-Clause License

Copyright (c) 2017-2019, Revar Desmera
All rights reserved.

Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are met:

* Redistributions of source code must retain the above copyright notice, this
  list of conditions and the following disclaimer.

* Redistributions in binary form must reproduce the above copyright notice,
  this list of conditions and the following disclaimer in the documentation
  and/or other materials provided with the distribution.

THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
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DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE
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SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
*/



// Section: Lines and Triangles

// Function: point_on_segment()
// Usage:
//   point_on_segment(point, edge);
// Description:
//   Determine if the point is on the line segment between two points.
//   Returns true if yes, and false if not.  
// Arguments:
//   point = The point to check colinearity of.
//   edge = Array of two points forming the line segment to test against.
function point_on_segment(point, edge) =
	point==edge[0] || point==edge[1] ||  // The point is an endpoint
	sign(edge[0].x-point.x)==sign(point.x-edge[1].x)  // point is in between the
		&& sign(edge[0].y-point.y)==sign(point.y-edge[1].y)  // edge endpoints 
		&& point_left_of_segment(point, edge)==0;  // and on the line defined by edge


// Function: point_left_of_segment()
// Usage:
//   point_left_of_segment(point, edge);
// Description:
//   Return >0 if point is left of the line defined by edge.
//   Return =0 if point is on the line.
//   Return <0 if point is right of the line.
// Arguments:
//   point = The point to check position of.
//   edge = Array of two points forming the line segment to test against.
function point_left_of_segment(point, edge) =
	(edge[1].x-edge[0].x) * (point.y-edge[0].y) - (point.x-edge[0].x) * (edge[1].y-edge[0].y);
  

// Internal non-exposed function.
function _point_above_below_segment(point, edge) =
	edge[0].y <= point.y? (
		(edge[1].y > point.y && point_left_of_segment(point, edge) > 0)? 1 : 0
	) : (
		(edge[1].y <= point.y && point_left_of_segment(point, edge) < 0)? -1 : 0
	);


// Function: right_of_line2d()
// Usage:
//   right_of_line2d(line, pt)
// Description:
//   Returns true if the given point is to the left of the given line.
// Arguments:
//   line = A list of two points.
//   pt = The point to test.
function right_of_line2d(line, pt) =
	triangle_area2d(line[0], line[1], pt) < 0;


// Function: collinear()
// Usage:
//   collinear(a, b, c, [eps]);
// Description:
//   Returns true if three points are co-linear.
// Arguments:
//   a = First point.
//   b = Second point.
//   c = Third point.
//   eps = Acceptable max angle variance.  Default: EPSILON (1e-9) degrees.
function collinear(a, b, c, eps=EPSILON) =
	abs(vector_angle(b-a,c-a)) < eps;


// Function: collinear_indexed()
// Usage:
//   collinear_indexed(points, a, b, c, [eps]);
// Description:
//   Returns true if three points are co-linear.
// Arguments:
//   points = A list of points.
//   a = Index in `points` of first point.
//   b = Index in `points` of second point.
//   c = Index in `points` of third point.
//   eps = Acceptable max angle variance.  Default: EPSILON (1e-9) degrees.
function collinear_indexed(points, a, b, c, eps=EPSILON) =
	let(
		p1=points[a],
		p2=points[b],
		p3=points[c]
	) abs(vector_angle(p2-p1,p3-p1)) < eps;


// Function: triangle_area2d()
// Usage:
//   triangle_area2d(a,b,c);
// Description:
//   Returns the area of a triangle formed between three vertices.
//   Result will be negative if the points are in clockwise order.
// Examples:
//   triangle_area2d([0,0], [5,10], [10,0]);  // Returns -50
//   triangle_area2d([10,0], [5,10], [0,0]);  // Returns 50
function triangle_area2d(a,b,c) =
	(
		a.x * (b.y - c.y) + 
		b.x * (c.y - a.y) + 
		c.x * (a.y - b.y)
	) / 2;


// Section: Planes

// Function: plane3pt()
// Usage:
//   plane3pt(p1, p2, p3);
// Description:
//   Generates the cartesian equation of a plane from three non-colinear points on the plane.
//   Returns [A,B,C,D] where Ax+By+Cz+D=0 is the equation of a plane.
// Arguments:
//   p1 = The first point on the plane.
//   p2 = The second point on the plane.
//   p3 = The third point on the plane.
function plane3pt(p1, p2, p3) =
	let(normal = normalize(cross(p3-p1, p2-p1))) concat(normal, [normal*p1]);


// Function: plane3pt_indexed()
// Usage:
//   plane3pt_indexed(points, i1, i2, i3);
// Description:
//   Given a list of points, and the indexes of three of those points,
//   generates the cartesian equation of a plane that those points all
//   lie on.  Requires that the three indexed points be non-collinear.
//   Returns [A,B,C,D] where Ax+By+Cz+D=0 is the equation of a plane.
// Arguments:
//   points = A list of points.
//   i1 = The index into `points` of the first point on the plane.
//   i2 = The index into `points` of the second point on the plane.
//   i3 = The index into `points` of the third point on the plane.
function plane3pt_indexed(points, i1, i2, i3) =
	let(
		p1 = points[i1],
		p2 = points[i2],
		p3 = points[i3],
		normal = normalize(cross(p3-p1, p2-p1))
	) concat(normal, [normal*p1]);


// Function: distance_from_plane()
// Usage:
//   distance_from_plane(plane, point)
// Description:
//   Given a plane as [A,B,C,D] where the cartesian equation for that plane
//   is Ax+By+Cz+D=0, determines how far from that plane the given point is.
//   The returned distance will be positive if the point is in front of the
//   plane; on the same side of the plane as the normal of that plane points
//   towards.  If the point is behind the plane, then the distance returned
//   will be negative.  The normal of the plane is the same as [A,B,C].
// Arguments:
//   plane = The [A,B,C,D] values for the equation of the plane.
//   point = The point to test.
function distance_from_plane(plane, point) =
	[plane.x, plane.y, plane.z] * point - plane[3];


// Function: coplanar()
// Usage:
//   coplanar(plane, point);
// Description:
//   Given a plane as [A,B,C,D] where the cartesian equation for that plane
//   is Ax+By+Cz+D=0, determines if the given point is on that plane.
//   Returns true if the point is on that plane.
// Arguments:
//   plane = The [A,B,C,D] values for the equation of the plane.
//   point = The point to test.
function coplanar(plane, point) =
	abs(distance_from_plane(plane, point)) <= EPSILON;


// Function: in_front_of_plane()
// Usage:
//   in_front_of_plane(plane, point);
// Description:
//   Given a plane as [A,B,C,D] where the cartesian equation for that plane
//   is Ax+By+Cz+D=0, determines if the given point is on the side of that
//   plane that the normal points towards.  The normal of the plane is the
//   same as [A,B,C].
// Arguments:
//   plane = The [A,B,C,D] values for the equation of the plane.
//   point = The point to test.
function in_front_of_plane(plane, point) =
	distance_from_plane(plane, point) > EPSILON;



// Section: Paths and Polygons


// Function: simplify_path()
// Description:
//   Takes a path and removes unnecessary collinear points.
// Usage:
//   simplify_path(path, [eps])
// Arguments:
//   path = A list of 2D path points.
//   eps = Largest angle variance allowed.  Default: EPSILON (1-e9) degrees.
function simplify_path(path, eps=EPSILON, _a=0, _b=2, _acc=[]) =
	(_b >= len(path))? concat([path[0]], _acc, [path[len(path)-1]]) :
	simplify_path(
		path, eps,
		(collinear_indexed(path, _a, _b-1, _b, eps=eps)? _a : _b-1),
		_b+1,
		(collinear_indexed(path, _a, _b-1, _b, eps=eps)? _acc : concat(_acc, [path[_b-1]]))
	);


// Function: simplify_path_indexed()
// Description:
//   Takes a list of points, and a path as a list of indexes into `points`,
//   and removes all path points that are unecessarily collinear.
// Usage:
//   simplify_path_indexed(path, eps)
// Arguments:
//   points = A list of points.
//   path = A list of indexes into `points` that forms a path.
//   eps = Largest angle variance allowed.  Default: EPSILON (1-e9) degrees.
function simplify_path_indexed(points, path, eps=EPSILON, _a=0, _b=2, _acc=[]) =
	(_b >= len(path))? concat([path[0]], _acc, [path[len(path)-1]]) :
	simplify_path_indexed(
		points, path, eps,
		(collinear_indexed(points, path[_a], path[_b-1], path[_b], eps=eps)? _a : _b-1),
		_b+1,
		(collinear_indexed(points, path[_a], path[_b-1], path[_b], eps=eps)? _acc : concat(_acc, [path[_b-1]]))
	);


// Function: point_in_polygon()
// Usage:
//   point_in_polygon(point, path)
// Description:
//   This function tests whether the given point is inside, outside or on the boundary of
//   the specified polygon using the Winding Number method.  (http://geomalgorithms.com/a03-_inclusion.html)
//   The polygon is given as a list of points, not including the repeated end point.
//   Returns -1 if the point is outside the polyon.
//   Returns 0 if the point is on the boundary.
//   Returns 1 if the point lies in the interior.
//   The polygon does not need to be simple: it can have self-intersections.
//   But the polygon cannot have holes (it must be simply connected).
//   Rounding error may give mixed results for points on or near the boundary.
// Arguments:
//   point = The point to check position of.
//   path = The list of 2D path points forming the perimeter of the polygon.
function point_in_polygon(point, path) =
	// Does the point lie on any edges?  If so return 0. 
	sum([for(i=[0:len(path)-1]) point_on_segment(point, select(path, i, i+1))?1:0])>0 ? 0 : 
	// Otherwise compute winding number and return 1 for interior, -1 for exterior
	sum([for(i=[0:len(path)-1]) _point_above_below_segment(point, select(path, i, i+1))]) != 0 ? 1 : -1;


// Function: pointlist_bounds()
// Usage:
//   pointlist_bounds(pts);
// Description:
//   Finds the bounds containing all the points in pts.
//   Returns [[minx, miny, minz], [maxx, maxy, maxz]]
// Arguments:
//   pts = List of points.
function pointlist_bounds(pts) = [
	[for (a=[0:2]) min([ for (x=pts) point3d(x)[a] ]) ],
	[for (a=[0:2]) max([ for (x=pts) point3d(x)[a] ]) ]
];


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