From 4315b8c0b6d0e046c6f59c00000cacea9dfa1015 Mon Sep 17 00:00:00 2001 From: Adrian Mariano Date: Sat, 6 Nov 2021 10:42:45 -0400 Subject: [PATCH] doc fixes --- regions.scad | 218 ++++++++++++++++++++++++++++++--------------------- 1 file changed, 130 insertions(+), 88 deletions(-) diff --git a/regions.scad b/regions.scad index 6f11175..6fc0771 100644 --- a/regions.scad +++ b/regions.scad @@ -1,9 +1,11 @@ ////////////////////////////////////////////////////////////////////// // LibFile: regions.scad -// This file provides 2D boolean geometry operations on paths, where you can -// compute the intersection or union of the shape defined by point lists, producing -// a new point list. Of course, boolean operations may produce shapes with multiple -// components. To handle that, we use "regions" which are defined as lists of paths. +// This file provides 2D boolean set operations on polygons, where you can +// compute, for example, the intersection or union of the shape defined by point lists, producing +// a new point list. Of course, such operations may produce shapes with multiple +// components. To handle that, we use "regions" which are defined by lists of polygons. +// In addition to set operations, you can calculate offsets, determine whether a point is in a +// region and you can decompose a region into parts. // Includes: // include ////////////////////////////////////////////////////////////////////// @@ -16,30 +18,32 @@ // Section: Regions // A region is a list of polygons meeting these conditions: // . -// - Every polygon on the list is simpel, meaning it does not intersect itself +// - Every polygon on the list is simple, meaning it does not intersect itself // - Two polygons on the list do not cross each other // - A vertex of one polygon never meets the edge of another one except at a vertex // . // Note that this means vertex-vertex touching between two polygons is acceptable // to define a region. Note, however, that regions with vertex-vertex contact usually -// cannot be rendered with CGAL. +// cannot be rendered with CGAL. See {{is_valid_region()}} for examples of valid regions and +// lists of polygons that are not regions. Note that {{is_region_simple()}} will identify +// regions with no polygon intersections at all, which should render successfully witih CGAL. // . // The actual geometry of the region is defined by XORing together -// all of the polygons on the list. This may sound obscure, but it simply means that nested +// all of the polygons in the list. This may sound obscure, but it simply means that nested // boundaries make rings in the obvious fashion, and non-nested shapes simply union together. -// Checking that a list of paths is a valid region, meaning that it satisfies all of the conditions +// Checking that a list of polygons is a valid region, meaning that it satisfies all of the conditions // above, can be a time consuming test, so it is not done automatically. It is your responsibility to ensure that your regions are -// compliant. You can construct regions by making a list of polygons, or by using -// boolean function operations such as union() or difference(), which all acccept paths, as +// compliant. You can construct regions by making a suitable list of polygons, or by using +// set operation function such as union() or difference(), which all acccept polygons, as // well as regions, as their inputs. And if you must you can clean up an ill-formed region using make_region(), -// which will break up self-intersecting paths and paths that cross each other. +// which will break up self-intersecting polygons and polygons that cross each other. // Function: is_region() // Usage: // is_region(x); // Description: -// Returns true if the given item looks like a region. A region is a list of non-crossing simple paths. This test just checks +// Returns true if the given item looks like a region. A region is a list of non-crossing simple polygons. This test just checks // that the argument is a list whose first entry is a path. function is_region(x) = is_list(x) && is_path(x.x); @@ -48,16 +52,16 @@ function is_region(x) = is_list(x) && is_path(x.x); // Usage: // bool = is_valid_region(region, [eps]); // Description: -// Returns true if the input is a valid region, meaning that it is a list of simple paths whose segments do not cross each other. +// Returns true if the input is a valid region, meaning that it is a list of simple polygons whose segments do not cross each other. // This test can be time consuming with regions that contain many points. -// It differs from `is_region()` which simply checks that the object appears to be a list of paths -// because it searches all the region paths for any self-intersections or intersections with each other. -// Will also return true if given a single simple path. Use {{make_region()}} to convert sets of self-intersecting polygons into +// It differs from `is_region()` which simply checks that the object is a list whose first entry is a path +// because it searches all the list polygons for any self-intersections or intersections with each other. +// Will also return true if given a single simple polygon. Use {{make_region()}} to convert sets of self-intersecting polygons into // a region. // Arguments: // region = region to check // eps = tolerance for geometric comparisons. Default: `EPSILON` = 1e-9 -// Example(2D,NoAxes): In all of the examples each path in the region appears in a different color. Two non-intersecting squares make a valid region. +// Example(2D,NoAxes): In all of the examples each polygon in the region appears in a different color. Two non-intersecting squares make a valid region. // region = [square(10), right(11,square(8))]; // rainbow(region)stroke($item, width=.2,closed=true); // back(11)text(is_valid_region(region) ? "region" : "non-region", size=2); @@ -73,23 +77,27 @@ function is_region(x) = is_list(x) && is_path(x.x); // object = [square(10), move([8,8], square(8))]; // rainbow(object)stroke($item, width=.2,closed=true); // back(17)text(is_valid_region(object) ? "region" : "non-region", size=2); -// Example(2D,NoAxes): Not a region due to a self-intersecting (non-simple) hourglass path +// Example(2D,NoAxes): A union is one way to fix the above example and get a region. (Note that union is run here on two simple polygons, which are valid regions themselves and hence acceptable inputs to union. +// region = union([square(10), move([8,8], square(8))]); +// rainbow(region)stroke($item, width=.25,closed=true); +// back(12)text(is_valid_region(region) ? "region" : "non-region", size=2); +// Example(2D,NoAxes): Not a region due to a self-intersecting (non-simple) hourglass polygon // object = [move([-2,-2],square(14)), [[0,0],[10,0],[0,10],[10,10]]]; // rainbow(object)stroke($item, width=.2,closed=true); // move([-1.5,13])text(is_valid_region(object) ? "region" : "non-region", size=2); // Example(2D,NoAxes): Breaking hourglass in half fixes it. Now it's a region: // region = [move([-2,-2],square(14)), [[0,0],[10,0],[5,5]], [[5,5],[0,10],[10,10]]]; // rainbow(region)stroke($item, width=.2,closed=true); -// Example(2D,NoAxes): A single path corner touches an edge, so not a region: +// Example(2D,NoAxes): A single polygon corner touches an edge, so not a region: // object = [[[-10,0], [-10,10], [20,10], [20,-20], [-10,-20], // [-10,-10], [0,0], [10,-10], [10,0]]]; // rainbow(object)stroke($item, width=.3,closed=true); // move([-4,12])text(is_valid_region(object) ? "region" : "non-region", size=3); -// Example(2D,NoAxes): Corners touch in the same path, so the path is not simple and the object is not a region. +// Example(2D,NoAxes): Corners touch in the same polygon, so the polygon is not simple and the object is not a region. // object = [[[0,0],[10,0],[10,10],[-10,10],[-10,0],[0,0],[-5,5],[5,5]]]; // rainbow(object)stroke($item, width=.3,closed=true); // move([-10,12])text(is_valid_region(object) ? "region" : "non-region", size=3); -// Example(2D,NoAxes): The shape above as a valid region with two paths: +// Example(2D,NoAxes): The shape above as a valid region with two polygons: // region = [ [[0,0],[10,0],[10,10],[-10,10],[-10,0]], // [[0,0],[5,5],[-5,5]] ]; // rainbow(region)stroke($item, width=.3,closed=true); @@ -98,10 +106,6 @@ function is_region(x) = is_list(x) && is_path(x.x); // region = [square(10), move([10,10], square(8))]; // rainbow(region)stroke($item, width=.25,closed=true); // back(12)text(is_valid_region(region) ? "region" : "non-region", size=2); -// Example(2D,NoAxes): A union is one way to fix the above example and get a region. (Note that union is run here on two simple paths, which are valid regions themselves and hence acceptable inputs to union. -// region = union([square(10), move([8,8], square(8))]); -// rainbow(region)stroke($item, width=.25,closed=true); -// back(12)text(is_valid_region(region) ? "region" : "non-region", size=2); // Example(2D,NoAxes): These two squares share part of an edge, hence not a region // object = [square(10), move([10,2], square(7))]; // stroke(object[0], width=0.2,closed=true); @@ -121,11 +125,11 @@ function is_region(x) = is_list(x) && is_path(x.x); // object = [square(10), [[10,0],[0,10],[8,13],[13,8]]]; // rainbow(object)stroke($item, width=.2,closed=true); // back(14)text(is_valid_region(object) ? "region" : "non-region", size=2); -// Example(2D,NoAxes): One path touches another in the middle of an edge +// Example(2D,NoAxes): One polygon touches another in the middle of an edge // object = [square(10), [[10,5],[15,0],[15,10]]]; // rainbow(object)stroke($item, width=.2,closed=true); // back(11)text(is_valid_region(object) ? "region" : "non-region", size=2); -// Example(2D,NoAxes): The path touches the side, but the side has a vertex at the contact point so this is a region +// Example(2D,NoAxes): The polygon touches the side, but the side has a vertex at the contact point so this is a region // poly1 = [ each square(30,center=true), [15,0]]; // poly2 = right(10,circle(5,$fn=4)); // poly3 = left(0,circle(5,$fn=4)); @@ -134,7 +138,7 @@ function is_region(x) = is_list(x) && is_path(x.x); // rainbow(region)stroke($item, width=.25,closed=true); // move([-5,16.5])text(is_valid_region(region) ? "region" : "non-region", size=3); // color("black")move_copies(region[0]) circle(r=.4); -// Example(2D,NoAxes): The path touches the side, but not at a vertex so this is not a region +// Example(2D,NoAxes): The polygon touches the side, but not at a vertex so this is not a region // poly1 = fwd(4,[ each square(30,center=true), [15,0]]); // poly2 = right(10,circle(5,$fn=4)); // poly3 = left(0,circle(5,$fn=4)); @@ -143,13 +147,13 @@ function is_region(x) = is_list(x) && is_path(x.x); // rainbow(object)stroke($item, width=.25,closed=true); // move([-9,12.5])text(is_valid_region(object) ? "region" : "non-region", size=3); // color("black")move_copies(object[0]) circle(r=.4); -// Example(2D,NoAxes): The inner path touches the middle of the edges, so not a region +// Example(2D,NoAxes): The inner polygon touches the middle of the edges, so not a region // poly1 = square(20,center=true); // poly2 = circle(10,$fn=8); // object=[poly1,poly2]; // rainbow(object)stroke($item, width=.25,closed=true); // move([-10,11.4])text(is_valid_region(object) ? "region" : "non-region", size=3); -// Example(2D,NoAxes): The above shape made into a region using difference(): +// Example(2D,NoAxes): The above shape made into a region using {{difference()}} now has four components that touch at corners // poly1 = square(20,center=true); // poly2 = circle(10,$fn=8); // region = difference(poly1,poly2); @@ -212,7 +216,15 @@ function _polygon_crosses_region(region, poly, eps=EPSILON) = // Arguments: // region = region to check // eps = tolerance for geometric comparisons. Default: `EPSILON` = 1e-9 -function is_simple_region(region, eps=EPSILON) = +// Example(2D,NoAxes): Corner contact means it's not simple +// region = [move([-2,-2],square(14)), [[0,0],[10,0],[5,5]], [[5,5],[0,10],[10,10]]]; +// rainbow(region)stroke($item, width=.2,closed=true); +// move([-1,13])text(is_region_simple(region) ? "simple" : "not-simple", size=2); +// Example(2D,NoAxes): Moving apart the triangles makes it simple: +// region = [move([-2,-2],square(14)), [[0,0],[10,0],[5,4.5]], [[5,5.5],[0,10],[10,10]]]; +// rainbow(region)stroke($item, width=.2,closed=true); +// move([1,13])text(is_region_simple(region) ? "simple" : "not-simple", size=2); +function is_region_simple(region, eps=EPSILON) = let(region=force_region(region)) assert(is_region(region), "Input is not a region") [for(p=region) if (!is_path_simple(p,closed=true,eps=eps)) 1] == [] @@ -224,32 +236,41 @@ function is_simple_region(region, eps=EPSILON) = // Function: make_region() // Usage: -// r_fixed = make_region(r, [nonzero], [eps]); +// region = make_region(polys, [nonzero], [eps]); // Description: -// Takes a malformed input region that contains self-intersecting polygons or polygons -// that cross each other and converts it into a properly defined region without +// Takes a list of polygons that may intersect themselves or cross each other +// and converts it into a properly defined region without // these defects. // Arguments: -// r = region to sanitize +// polys = list of polygons to use // nonzero = set to true to use nonzero rule for polygon membership. Default: false // eps = Epsilon for geometric comparisons. Default: `EPSILON` (1e-9) -// Examples: -// -function make_region(r,nonzero=false,eps=EPSILON) = - let(r=force_region(r)) - assert(is_region(r), "Input is not a region") +// Example(2D,NoAxes): The pentagram is self-intersecting, so it is not a region. Here it becomes five triangles: +// pentagram = turtle(["move",100,"left",144], repeat=4); +// region = make_region(pentagram); +// rainbow(region)stroke($item, width=1,closed=true); +// Examples(2D,NoAxes): Alternatively with the nonzero option you can get the perimeter: +// pentagram = turtle(["move",100,"left",144], repeat=4); +// region = make_region(pentagram,nonzero=true); +// stroke(region, width=1,closed=true); +// Example(2D,NoAxes): To crossing squares become two L-shaped components +// region = make_region([square(10), move([5,5],square(8))]); +// rainbow(region)stroke($item, width=.3,closed=true); +function make_region(polys,nonzero=false,eps=EPSILON) = + let(polys=force_region(polys)) + assert(is_region(polys), "Input is not a region") exclusive_or( - [for(poly=r) each polygon_parts(poly,nonzero,eps)], + [for(poly=polys) each polygon_parts(poly,nonzero,eps)], eps=eps); - - // Function: force_region() // Usage: -// region = force_region(path) +// region = force_region(poly) // Description: -// If the input is a path then return it as a region. Otherwise return it unaltered. -function force_region(path) = is_path(path) ? [path] : path; +// If the input is a polygon then return it as a region. Otherwise return it unaltered. +// Arguments: +// poly = polygon to turn into a region +function force_region(poly) = is_path(poly) ? [poly] : poly; // Section: Turning a region into geometry @@ -258,12 +279,14 @@ function force_region(path) = is_path(path) ? [path] : path; // Usage: // region(r); // Description: -// Creates 2D polygons for the given region. The region given is a list of closed 2D paths. -// Each path will be effectively exclusive-ORed from all other paths in the region, so if a -// path is inside another path, it will be effectively subtracted from it. -// Example(2D): +// Creates the 2D polygons described by the given region or list of polygons. This module works on +// arbitrary lists of polygons that cross each other and hence do not define a valid region. The +// displayed result is the exclusive-or of the polygons listed in the input. +// Arguments: +// r = region to create as geometry +// Example(2D): Displaying a region // region([circle(d=50), square(25,center=true)]); -// Example(2D): +// Example(2D): Displaying a list of polygons that intersect each other, which is not a region // rgn = concat( // [for (d=[50:-10:10]) circle(d=d-5)], // [square([60,10], center=true)] @@ -283,27 +306,36 @@ module region(r) -// Section: Gometrical calculations with region +// Section: Gometrical calculations with regions // Function: point_in_region() // Usage: // check = point_in_region(point, region, [eps]); // Description: -// Tests if a point is inside, outside, or on the border of a region. +// Tests if a point is inside, outside, or on the border of a region. // Returns -1 if the point is outside the region. // Returns 0 if the point is on the boundary. // Returns 1 if the point lies inside the region. // Arguments: // point = The point to test. -// region = The region to test against. Given as a list of polygon paths. +// region = The region to test against, as a list of polygon paths. // eps = Acceptable variance. Default: `EPSILON` (1e-9) -function point_in_region(point, region, eps=EPSILON, _i=0, _cnt=0) = - _i >= len(region) ? ((_cnt%2==1)? 1 : -1) - : let( - pip = point_in_polygon(point, region[_i], eps=eps) - ) - pip==0? 0 - : point_in_region(point, region, eps=eps, _i=_i+1, _cnt = _cnt + (pip>0? 1 : 0)); +// Example(2D,NoAxes): +// region = [for(i=[2:8]) hexagon(r=i)]; +// for(x=[-4:4],y=[-4:4]) color(point_in_region +function point_in_region(point, region, eps=EPSILON) = + let(region=force_region(region)) + assert(is_region(region), "Region given to point_in_region is not a region") + assert(is_vector(point,2), "Point must be a 2D point in point_in_region") + _point_in_region(point, region, eps); + +function pointin_region(point, region, eps=EPSILON, i=0, cnt=0) = + i >= len(region) ? ((cnt%2==1)? 1 : -1) + : let( + pip = point_in_polygon(point, region[i], eps=eps) + ) + pip == 0 ? 0 + : _point_in_region(point, region, eps=eps, i=i+1, cnt = cnt + (pip>0? 1 : 0)); // Function: region_area() @@ -312,6 +344,10 @@ function point_in_region(point, region, eps=EPSILON, _i=0, _cnt=0) = // Description: // Computes the area of the specified valid region. (If the region is invalid and has self intersections // the result is meaningless.) +// Arguments: +// region = region whose area to compute +// Examples: +// area = region_area([square(10), right(20,square(8))]); // Returns 164 function region_area(region) = assert(is_region(region), "Input must be a region") let( @@ -437,9 +473,13 @@ function _region_region_intersections(region1, region2, closed1=true,closed2=tru // Splits region1 at the places where polygons in region1 touches each other at corners and at locations // where region1 intersections region2. Split region2 similarly with respect to region1. // The return is a pair of results of the form [split1, split2] where split1=[frags1,frags2,...] -// and frags1 is a list of path pieces (in order) from the first path of the region. -// You can pass a single path in for either region, but the output will be a singleton list, as ify -// you passed in a singleton region. +// and frags1 is a list of paths that when placed end to end (in the given order), give the first polygon of region1. +// Each path in the list is either entirely inside or entirely outside region2. +// Then frags2 is the decomposition of the second polygon into path pieces, and so on. Finally split2 is +// the same list, but for the polygons in region2. +// You can pass a single polygon in for either region, but the output will be a singleton list, as if +// you passed in a singleton region. If you set the closed parameters to false then the region components +// will be treated as open paths instead of polygons. // Arguments: // region1 = first region // region2 = second region @@ -543,9 +583,9 @@ function region_parts(region) = // Usage: // linear_sweep(region, height, [center], [slices], [twist], [scale], [style], [convexity]); // Description: -// If called as a module, creates a polyhedron that is the linear extrusion of the given 2D region or path. +// If called as a module, creates a polyhedron that is the linear extrusion of the given 2D region or polygon. // If called as a function, returns a VNF that can be used to generate a polyhedron of the linear extrusion -// of the given 2D region or path. The benefit of using this, over using `linear_extrude region(rgn)` is +// of the given 2D region or polygon. The benefit of using this, over using `linear_extrude region(rgn)` is // that it supports `anchor`, `spin`, `orient` and attachments. You can also make more refined // twisted extrusions by using `maxseg` to subsample flat faces. // Note that the center option centers vertically using the named anchor "zcenter" whereas @@ -553,7 +593,7 @@ function region_parts(region) = // the shape's centroid, or other centerpoint you specify. The centerpoint can be "centroid", "mean", "box" or // a custom point location. // Arguments: -// region = The 2D [Region](regions.scad) or path that is to be extruded. +// region = The 2D [Region](regions.scad) or polygon that is to be extruded. // height = The height to extrude the region. Default: 1 // center = If true, the created polyhedron will be vertically centered. If false, it will be extruded upwards from the XY plane. Default: `false` // slices = The number of slices to divide the shape into along the Z axis, to allow refinement of detail, especially when working with a twist. Default: `twist/5` @@ -656,7 +696,7 @@ function linear_sweep(region, height=1, center, twist=0, scale=1, slices, -// Section: Offsets and Boolean 2D Geometry +// Section: Offset and 2D Boolean Set Operations function _offset_chamfer(center, points, delta) = @@ -773,10 +813,11 @@ function _point_dist(path,pathseg_unit,pathseg_len,pt) = // offsetpath = offset(path, [r|delta], [chamfer], [closed], [check_valid], [quality]) // path_faces = offset(path, return_faces=true, [r|delta], [chamfer], [closed], [check_valid], [quality], [firstface_index], [flip_faces]) // Description: -// Takes a 2D input path and returns a path offset by the specified amount. As with the built-in +// Takes a 2D input path, polygon or region and returns a path offset by the specified amount. As with the built-in // offset() module, you can use `r` to specify rounded offset and `delta` to specify offset with // corners. If you used `delta` you can set `chamfer` to true to get chamfers. -// Positive offsets shift the path to the left (relative to the direction of the path). Note +// For paths and polygons positive offsets make the polygons larger. For paths, +// positive offsets shift the path to the left, relative to the direction of the path. Note // that the path must not include any 180 degree turns, where the path reverses direction. // . // When offsets shrink the path, segments cross and become invalid. By default `offset()` checks @@ -800,7 +841,7 @@ function _point_dist(path,pathseg_unit,pathseg_len,pt) = // r = offset radius. Distance to offset. Will round over corners. // delta = offset distance. Distance to offset with pointed corners. // chamfer = chamfer corners when you specify `delta`. Default: false -// closed = path is a closed curve. Default: False. +// closed = if true path is treate as a polygon. Default: False. // check_valid = perform segment validity check. Default: True. // quality = validity check quality parameter, a small integer. Default: 1. // return_faces = return face list. Default: False. @@ -1050,11 +1091,11 @@ function _list_three(a,b,c) = // region = union(REGION1,REGION2); // region = union(REGION1,REGION2,REGION3); // Description: -// When called as a function and given a list of regions, where each region is a list of closed -// 2D paths, returns the boolean union of all given regions. Result is a single region. -// When called as the built-in module, makes the boolean union of the given children. +// When called as a function and given a list of regions or 2D polygons, +// returns the union of all given regions and polygons. Result is a single region. +// When called as the built-in module, makes the union of the given children. // Arguments: -// regions = List of regions to union. Each region is a list of closed paths. +// regions = List of regions to union. // Example(2D): // shape1 = move([-8,-8,0], p=circle(d=50)); // shape2 = move([ 8, 8,0], p=circle(d=50)); @@ -1080,12 +1121,12 @@ function union(regions=[],b=undef,c=undef,eps=EPSILON) = // region = difference(REGION1,REGION2); // region = difference(REGION1,REGION2,REGION3); // Description: -// When called as a function, and given a list of regions, where each region is a list of closed -// 2D paths, takes the first region and differences away all other regions from it. The resulting +// When called as a function, and given a list of regions or 2D polygons, +// takes the first region or polygon and differences away all other regions/polygons from it. The resulting // region is returned. -// When called as the built-in module, makes the boolean difference of the given children. +// When called as the built-in module, makes the set difference of the given children. // Arguments: -// regions = List of regions to difference. Each region is a list of closed paths. +// regions = List of regions or polygons to difference. // Example(2D): // shape1 = move([-8,-8,0], p=circle(d=50)); // shape2 = move([ 8, 8,0], p=circle(d=50)); @@ -1112,11 +1153,11 @@ function difference(regions=[],b=undef,c=undef,eps=EPSILON) = // region = intersection(REGION1,REGION2); // region = intersection(REGION1,REGION2,REGION3); // Description: -// When called as a function, and given a list of regions, where each region is a list of closed -// 2D paths, returns the boolean intersection of all given regions. Result is a single region. -// When called as the built-in module, makes the boolean intersection of all the given children. +// When called as a function, and given a list of regions or polygons returns the +// intersection of all given regions. Result is a single region. +// When called as the built-in module, makes the intersection of all the given children. // Arguments: -// regions = List of regions to intersection. Each region is a list of closed paths. +// regions = List of regions to intersect. // Example(2D): // shape1 = move([-8,-8,0], p=circle(d=50)); // shape2 = move([ 8, 8,0], p=circle(d=50)); @@ -1143,12 +1184,13 @@ function intersection(regions=[],b=undef,c=undef,eps=EPSILON) = // region = exclusive_or(REGION1,REGION2); // region = exclusive_or(REGION1,REGION2,REGION3); // Description: -// When called as a function and given a list of regions, where each region is a list of closed -// 2D paths, returns the boolean exclusive_or of all given regions. Result is a single region. -// When called as a module, performs a boolean exclusive-or of up to 10 children. Note that the -// xor operator tends to produce shapes that meet at corners, which do not render in CGAL. +// When called as a function and given a list of regions or 2D polygons, +// returns the exclusive_or of all given regions. Result is a single region. +// When called as a module, performs a boolean exclusive-or of up to 10 children. Note that when +// the input regions cross each other the exclusive-or operator will produce shapes that +// meet at corners (non-simple regions), which do not render in CGAL. // Arguments: -// regions = List of regions to exclusive_or. Each region is a list of closed paths. +// regions = List of regions or polygons to exclusive_or // Example(2D): As Function. A linear_sweep of this shape fails to render in CGAL. // shape1 = move([-8,-8,0], p=circle(d=50)); // shape2 = move([ 8, 8,0], p=circle(d=50));