From 7b5c2456ced0d7232b19e19cf6f560c4c4e55e87 Mon Sep 17 00:00:00 2001
From: Adrian Mariano <avm4@cornell.edu>
Date: Tue, 28 Jul 2020 17:54:57 -0400
Subject: [PATCH] change hull to use find_noncollinear_points, add some error
 checks, add regression tests

---
 hull.scad            | 498 ++++++++++++++++++++++---------------------
 tests/test_hull.scad | 100 +++++++++
 2 files changed, 357 insertions(+), 241 deletions(-)
 create mode 100644 tests/test_hull.scad

diff --git a/hull.scad b/hull.scad
index 1855ce0..da7d2c9 100644
--- a/hull.scad
+++ b/hull.scad
@@ -1,241 +1,257 @@
-//////////////////////////////////////////////////////////////////////
-// LibFile: hull.scad
-//   Functions to create 2D and 3D convex hulls.
-//   To use, add the following line to the beginning of your file:
-//   ```
-//   include <BOSL2/std.scad>
-//   include <BOSL2/hull.scad>
-//   ```
-//   Derived from Oskar Linde's Hull:
-//   - https://github.com/openscad/scad-utils
-//////////////////////////////////////////////////////////////////////
-
-
-// Section: Convex Hulls
-
-
-// Function: hull()
-// Usage:
-//   hull(points);
-// Description:
-//   Takes a list of 2D or 3D points (but not both in the same list) and returns either the list of
-//   indexes into `points` that forms the 2D convex hull perimeter path, or the list of faces that
-//   form the 3d convex hull surface.  Each face is a list of indexes into `points`.  If the input
-//   points are co-linear, the result will be the indexes of the two extrema points.  If the input
-//   points are co-planar, the results will be a simple list of vertex indices that will form a planar
-//   perimeter.  Otherwise a list of faces will be returned, where each face is a simple list of
-//   vertex indices for the perimeter of the face.
-// Arguments:
-//   points = The set of 2D or 3D points to find the hull of.
-function hull(points) = let(two_d = len(points[0]) == 2) two_d? hull2d_path(points) : hull3d_faces(points);
-
-
-// Module: hull_points()
-// Usage:
-//   hull_points(points, [fast]);
-// Description:
-//   If given a list of 2D points, creates a 2D convex hull polygon that encloses all those points.
-//   If given a list of 3D points, creates a 3D polyhedron that encloses all the points.  This should
-//   handle about 4000 points in slow mode.  If `fast` is set to true, this should be able to handle
-//   far more.
-// Arguments:
-//   points = The list of points to form a hull around.
-//   fast = If true, uses a faster cheat that may handle more points, but also may emit warnings that can stop your script if you have "Halt on first warning" enabled.  Default: false
-// Example(2D):
-//   pts = [[-10,-10], [0,10], [10,10], [12,-10]];
-//   hull_points(pts);
-// Example:
-//   pts = [for (phi = [30:60:150], theta = [0:60:359]) spherical_to_xyz(10, theta, phi)];
-//   hull_points(pts);
-module hull_points(points, fast=false) {
-    assert(is_list(points));
-    if (points) {
-        assert(is_list(points[0]));
-        if (fast) {
-            if (len(points[0]) == 2) {
-                hull() polygon(points=points);
-            } else {
-                extra = len(points)%3;
-                faces = concat(
-                    [[for(i=[0:1:extra+2])i]],
-                    [for(i=[extra+3:3:len(points)-3])[i,i+1,i+2]]
-                );
-                hull() polyhedron(points=points, faces=faces);
-            }
-        } else {
-            perim = hull(points);
-            if (is_num(perim[0])) {
-                polygon(points=points, paths=[perim]);
-            } else {
-                polyhedron(points=points, faces=perim);
-            }
-        }
-    }
-}
-
-
-// Function: hull2d_path()
-// Usage:
-//   hull2d_path(points)
-// Description:
-//   Takes a list of arbitrary 2D points, and finds the minimal convex hull polygon to enclose them.
-//   Returns a path as a list of indices into `points`.
-// Example(2D):
-//   pts = [[-10,-10], [0,10], [10,10], [12,-10]];
-//   path = hull2d_path(pts);
-//   move_copies(pts) color("red") sphere(1);
-//   polygon(points=pts, paths=[path]);
-function hull2d_path(points) =
-    (len(points) < 3)? [] : let(
-        a=0, b=1,
-        c = first_noncollinear(a, b, points)
-    ) (c == len(points))? _hull2d_collinear(points) : let(
-        remaining = [ for (i = [2:1:len(points)-1]) if (i != c) i ],
-        ccw = triangle_area(points[a], points[b], points[c]) > 0,
-        polygon = ccw? [a,b,c] : [a,c,b]
-    ) _hull2d_iterative(points, polygon, remaining);
-
-
-// Adds the remaining points one by one to the convex hull
-function _hull2d_iterative(points, polygon, remaining, _i=0) =
-    (_i >= len(remaining))? polygon : let (
-        // pick a point
-        i = remaining[_i],
-        // find the segments that are in conflict with the point (point not inside)
-        conflicts = _find_conflicting_segments(points, polygon, points[i])
-        // no conflicts, skip point and move on
-    ) (len(conflicts) == 0)? _hull2d_iterative(points, polygon, remaining, _i+1) : let(
-        // find the first conflicting segment and the first not conflicting
-        // conflict will be sorted, if not wrapping around, do it the easy way
-        polygon = _remove_conflicts_and_insert_point(polygon, conflicts, i)
-    ) _hull2d_iterative(points, polygon, remaining, _i+1);
-
-
-function _hull2d_collinear(points) =
-    let(
-        a = points[0],
-        n = points[1] - a,
-        points1d = [ for(p = points) (p-a)*n ],
-        min_i = min_index(points1d),
-        max_i = max_index(points1d)
-    ) [min_i, max_i];
-
-
-function _find_conflicting_segments(points, polygon, point) = [
-    for (i = [0:1:len(polygon)-1]) let(
-        j = (i+1) % len(polygon),
-        p1 = points[polygon[i]],
-        p2 = points[polygon[j]],
-        area = triangle_area(p1, p2, point)
-    ) if (area < 0) i
-];
-
-
-// remove the conflicting segments from the polygon
-function _remove_conflicts_and_insert_point(polygon, conflicts, point) = 
-    (conflicts[0] == 0)? let(
-        nonconflicting = [ for(i = [0:1:len(polygon)-1]) if (!in_list(i, conflicts)) i ],
-        new_indices = concat(nonconflicting, (nonconflicting[len(nonconflicting)-1]+1) % len(polygon)),
-        polygon = concat([ for (i = new_indices) polygon[i] ], point)
-    ) polygon : let(
-        before_conflicts = [ for(i = [0:1:min(conflicts)]) polygon[i] ],
-        after_conflicts  = (max(conflicts) >= (len(polygon)-1))? [] : [ for(i = [max(conflicts)+1:1:len(polygon)-1]) polygon[i] ],
-        polygon = concat(before_conflicts, point, after_conflicts)
-    ) polygon;
-
-
-
-// Function: hull3d_faces()
-// Usage:
-//   hull3d_faces(points)
-// Description:
-//   Takes a list of arbitrary 3D points, and finds the minimal convex hull polyhedron to enclose
-//   them.  Returns a list of faces, where each face is a list of indexes into the given `points`
-//   list.  If all points passed to it are coplanar, then the return is the list of indices of points
-//   forming the minimal convex hull polygon.
-// Example(3D):
-//   pts = [[-20,-20,0], [20,-20,0], [0,20,5], [0,0,20]];
-//   faces = hull3d_faces(pts);
-//   move_copies(pts) color("red") sphere(1);
-//   %polyhedron(points=pts, faces=faces);
-function hull3d_faces(points) = 
-    (len(points) < 3)? list_range(len(points)) : let (    
-        // start with a single non-collinear triangle
-        a = 0,
-        b = 1,
-        c = first_noncollinear(a, b, points)
-    ) (c == len(points))? _hull2d_collinear(points) : let(
-        plane = plane3pt_indexed(points, a, b, c),
-        d = _find_first_noncoplanar(plane, points, 3)
-    ) (d == len(points))? /* all coplanar*/ let (
-        pts2d = [ for (p = points) project_plane(p, points[a], points[b], points[c]) ],
-        hull2d = hull2d_path(pts2d)
-    ) hull2d : let(
-        remaining = [for (i = [3:1:len(points)-1]) if (i != d) i],
-        // Build an initial tetrahedron.
-        // Swap b, c if d is in front of triangle t.
-        ifop = in_front_of_plane(plane, points[d]),
-        bc = ifop? [c,b] : [b,c],
-        b = bc[0],
-        c = bc[1],
-        triangles = [
-            [a,b,c],
-            [d,b,a],
-            [c,d,a],
-            [b,d,c]
-        ],
-        // calculate the plane equations
-        planes = [ for (t = triangles) plane3pt_indexed(points, t[0], t[1], t[2]) ]
-    ) _hull3d_iterative(points, triangles, planes, remaining);
-
-
-// Adds the remaining points one by one to the convex hull
-function _hull3d_iterative(points, triangles, planes, remaining, _i=0) =
-    _i >= len(remaining) ? triangles : 
-    let (
-        // pick a point
-        i = remaining[_i],
-        // find the triangles that are in conflict with the point (point not inside)
-        conflicts = _find_conflicts(points[i], planes),
-        // for all triangles that are in conflict, collect their halfedges
-        halfedges = [ 
-            for(c = conflicts, i = [0:2]) let(
-                j = (i+1)%3
-            ) [triangles[c][i], triangles[c][j]]
-        ],
-        // find the outer perimeter of the set of conflicting triangles
-        horizon = _remove_internal_edges(halfedges),
-        // generate a new triangle for each horizon halfedge together with the picked point i
-        new_triangles = [ for (h = horizon) concat(h,i) ],
-        // calculate the corresponding plane equations
-        new_planes = [ for (t = new_triangles) plane3pt_indexed(points, t[0], t[1], t[2]) ]
-    ) _hull3d_iterative(
-        points,
-        //  remove the conflicting triangles and add the new ones
-        concat(list_remove(triangles, conflicts), new_triangles),
-        concat(list_remove(planes, conflicts), new_planes),
-        remaining,
-        _i+1
-    );
-
-
-function _remove_internal_edges(halfedges) = [
-    for (h = halfedges)
-        if (!in_list(reverse(h), halfedges))
-            h
-];
-
-
-function _find_conflicts(point, planes) = [
-    for (i = [0:1:len(planes)-1])
-        if (in_front_of_plane(planes[i], point))
-            i
-];
-
-
-function _find_first_noncoplanar(plane, points, i) = 
-    (i >= len(points) || !coplanar(plane, points[i]))? i :
-    _find_first_noncoplanar(plane, points, i+1);
-
-
-// vim: expandtab tabstop=4 shiftwidth=4 softtabstop=4 nowrap
+//////////////////////////////////////////////////////////////////////
+// LibFile: hull.scad
+//   Functions to create 2D and 3D convex hulls.
+//   To use, add the following line to the beginning of your file:
+//   ```
+//   include <BOSL2/std.scad>
+//   include <BOSL2/hull.scad>
+//   ```
+//   Derived from Oskar Linde's Hull:
+//   - https://github.com/openscad/scad-utils
+//////////////////////////////////////////////////////////////////////
+
+
+// Section: Convex Hulls
+
+
+// Function: hull()
+// Usage:
+//   hull(points);
+// Description:
+//   Takes a list of 2D or 3D points (but not both in the same list) and returns either the list of
+//   indexes into `points` that forms the 2D convex hull perimeter path, or the list of faces that
+//   form the 3d convex hull surface.  Each face is a list of indexes into `points`.  If the input
+//   points are co-linear, the result will be the indexes of the two extrema points.  If the input
+//   points are co-planar, the results will be a simple list of vertex indices that will form a planar
+//   perimeter.  Otherwise a list of faces will be returned, where each face is a simple list of
+//   vertex indices for the perimeter of the face.
+// Arguments:
+//   points = The set of 2D or 3D points to find the hull of.
+function hull(points) =
+    assert(is_path(points),"Invalid input to hull")
+    len(points[0]) == 2
+      ? hull2d_path(points)
+      : hull3d_faces(points);
+
+
+// Module: hull_points()
+// Usage:
+//   hull_points(points, [fast]);
+// Description:
+//   If given a list of 2D points, creates a 2D convex hull polygon that encloses all those points.
+//   If given a list of 3D points, creates a 3D polyhedron that encloses all the points.  This should
+//   handle about 4000 points in slow mode.  If `fast` is set to true, this should be able to handle
+//   far more.
+// Arguments:
+//   points = The list of points to form a hull around.
+//   fast = If true, uses a faster cheat that may handle more points, but also may emit warnings that can stop your script if you have "Halt on first warning" enabled.  Default: false
+// Example(2D):
+//   pts = [[-10,-10], [0,10], [10,10], [12,-10]];
+//   hull_points(pts);
+// Example:
+//   pts = [for (phi = [30:60:150], theta = [0:60:359]) spherical_to_xyz(10, theta, phi)];
+//   hull_points(pts);
+module hull_points(points, fast=false) {
+    if (points) {
+        assert(is_list(points[0]));
+        if (fast) {
+            if (len(points[0]) == 2) {
+                hull() polygon(points=points);
+            } else {
+                extra = len(points)%3;
+                faces = concat(
+                    [[for(i=[0:1:extra+2])i]],
+                    [for(i=[extra+3:3:len(points)-3])[i,i+1,i+2]]
+                );
+                hull() polyhedron(points=points, faces=faces);
+            }
+        } else {
+            perim = hull(points);
+            if (is_num(perim[0])) {
+                polygon(points=points, paths=[perim]);
+            } else {
+                polyhedron(points=points, faces=perim);
+            }
+        }
+    }
+}
+
+
+// Function: hull2d_path()
+// Usage:
+//   hull2d_path(points)
+// Description:
+//   Takes a list of arbitrary 2D points, and finds the convex hull polygon to enclose them.
+//   Returns a path as a list of indices into `points`.  May return extra points, that are on edges of the hull.
+// Example(2D):
+//   pts = [[-10,-10], [0,10], [10,10], [12,-10]];
+//   path = hull2d_path(pts);
+//   move_copies(pts) color("red") sphere(1);
+//   polygon(points=pts, paths=[path]);
+function hull2d_path(points) =
+    assert(is_path(points,2),"Invalid input to hull2d_path")
+    len(points) < 2 ? []
+  : len(points) == 2 ? [0,1]
+  : let(tri=find_noncollinear_points(points, error=false))
+    tri == [] ? _hull_collinear(points)
+  : let(
+        remaining = [ for (i = [0:1:len(points)-1]) if (i != tri[0] && i!=tri[1] && i!=tri[2]) i ],
+        ccw = triangle_area(points[tri[0]], points[tri[1]], points[tri[2]]) > 0,
+        polygon = ccw ? [tri[0],tri[1],tri[2]] : [tri[0],tri[2],tri[1]]
+    ) _hull2d_iterative(points, polygon, remaining);
+
+
+
+// Adds the remaining points one by one to the convex hull
+function _hull2d_iterative(points, polygon, remaining, _i=0) =
+    (_i >= len(remaining))? polygon : let (
+        // pick a point
+        i = remaining[_i],
+        // find the segments that are in conflict with the point (point not inside)
+        conflicts = _find_conflicting_segments(points, polygon, points[i])
+        // no conflicts, skip point and move on
+    ) (len(conflicts) == 0)? _hull2d_iterative(points, polygon, remaining, _i+1) : let(
+        // find the first conflicting segment and the first not conflicting
+        // conflict will be sorted, if not wrapping around, do it the easy way
+        polygon = _remove_conflicts_and_insert_point(polygon, conflicts, i)
+    ) _hull2d_iterative(points, polygon, remaining, _i+1);
+
+
+function _hull_collinear(points) =
+    let(
+        a = points[0],
+        n = points[1] - a,
+        points1d = [ for(p = points) (p-a)*n ],
+        min_i = min_index(points1d),
+        max_i = max_index(points1d)
+    ) [min_i, max_i];
+
+
+function _find_conflicting_segments(points, polygon, point) = [
+    for (i = [0:1:len(polygon)-1]) let(
+        j = (i+1) % len(polygon),
+        p1 = points[polygon[i]],
+        p2 = points[polygon[j]],
+        area = triangle_area(p1, p2, point)
+    ) if (area < 0) i
+];
+
+
+// remove the conflicting segments from the polygon
+function _remove_conflicts_and_insert_point(polygon, conflicts, point) = 
+    (conflicts[0] == 0)? let(
+        nonconflicting = [ for(i = [0:1:len(polygon)-1]) if (!in_list(i, conflicts)) i ],
+        new_indices = concat(nonconflicting, (nonconflicting[len(nonconflicting)-1]+1) % len(polygon)),
+        polygon = concat([ for (i = new_indices) polygon[i] ], point)
+    ) polygon : let(
+        before_conflicts = [ for(i = [0:1:min(conflicts)]) polygon[i] ],
+        after_conflicts  = (max(conflicts) >= (len(polygon)-1))? [] : [ for(i = [max(conflicts)+1:1:len(polygon)-1]) polygon[i] ],
+        polygon = concat(before_conflicts, point, after_conflicts)
+    ) polygon;
+
+
+
+// Function: hull3d_faces()
+// Usage:
+//   hull3d_faces(points)
+// Description:
+//   Takes a list of arbitrary 3D points, and finds the convex hull polyhedron to enclose
+//   them.  Returns a list of triangular faces, where each face is a list of indexes into the given `points`
+//   list.  The output will be valid for use with the polyhedron command, but may include vertices that are in the interior of a face of the hull, so it is not
+//   necessarily the minimal representation of the hull.  
+//   If all points passed to it are coplanar, then the return is the list of indices of points
+//   forming the convex hull polygon.
+// Example(3D):
+//   pts = [[-20,-20,0], [20,-20,0], [0,20,5], [0,0,20]];
+//   faces = hull3d_faces(pts);
+//   move_copies(pts) color("red") sphere(1);
+//   %polyhedron(points=pts, faces=faces);
+function hull3d_faces(points) =
+    assert(is_path(points,3),"Invalid input to hull3d_faces")
+    len(points) < 3 ? list_range(len(points))
+  : let ( // start with a single non-collinear triangle
+          tri = find_noncollinear_points(points, error=false)
+        )
+    tri==[] ? _hull_collinear(points)
+  : let(
+        a = tri[0],
+        b = tri[1],
+        c = tri[2],
+        plane = plane3pt_indexed(points, a, b, c),
+        d = _find_first_noncoplanar(plane, points, 3)
+    )
+    d == len(points)
+  ? /* all coplanar*/
+    let (
+        pts2d = [ for (p = points) project_plane(p, points[a], points[b], points[c]) ],
+        hull2d = hull2d_path(pts2d)
+    ) hull2d
+  : let(
+        remaining = [for (i = [0:1:len(points)-1]) if (i!=a && i!=b && i!=c && i!=d) i],
+        // Build an initial tetrahedron.
+        // Swap b, c if d is in front of triangle t.
+        ifop = in_front_of_plane(plane, points[d]),
+        bc = ifop? [c,b] : [b,c],
+        b = bc[0],
+        c = bc[1],
+        triangles = [
+            [a,b,c],
+            [d,b,a],
+            [c,d,a],
+            [b,d,c]
+        ],
+        // calculate the plane equations
+        planes = [ for (t = triangles) plane3pt_indexed(points, t[0], t[1], t[2]) ]
+    ) _hull3d_iterative(points, triangles, planes, remaining);
+
+
+// Adds the remaining points one by one to the convex hull
+function _hull3d_iterative(points, triangles, planes, remaining, _i=0) =
+    _i >= len(remaining) ? triangles : 
+    let (
+        // pick a point
+        i = remaining[_i],
+        // find the triangles that are in conflict with the point (point not inside)
+        conflicts = _find_conflicts(points[i], planes),
+        // for all triangles that are in conflict, collect their halfedges
+        halfedges = [ 
+            for(c = conflicts, i = [0:2]) let(
+                j = (i+1)%3
+            ) [triangles[c][i], triangles[c][j]]
+        ],
+        // find the outer perimeter of the set of conflicting triangles
+        horizon = _remove_internal_edges(halfedges),
+        // generate a new triangle for each horizon halfedge together with the picked point i
+        new_triangles = [ for (h = horizon) concat(h,i) ],
+        // calculate the corresponding plane equations
+        new_planes = [ for (t = new_triangles) plane3pt_indexed(points, t[0], t[1], t[2]) ]
+    ) _hull3d_iterative(
+        points,
+        //  remove the conflicting triangles and add the new ones
+        concat(list_remove(triangles, conflicts), new_triangles),
+        concat(list_remove(planes, conflicts), new_planes),
+        remaining,
+        _i+1
+    );
+
+
+function _remove_internal_edges(halfedges) = [
+    for (h = halfedges)
+        if (!in_list(reverse(h), halfedges))
+            h
+];
+
+
+function _find_conflicts(point, planes) = [
+    for (i = [0:1:len(planes)-1])
+        if (in_front_of_plane(planes[i], point))
+            i
+];
+
+
+function _find_first_noncoplanar(plane, points, i) = 
+    (i >= len(points) || !coplanar(plane, points[i]))? i :
+    _find_first_noncoplanar(plane, points, i+1);
+
+
+// vim: expandtab tabstop=4 shiftwidth=4 softtabstop=4 nowrap
diff --git a/tests/test_hull.scad b/tests/test_hull.scad
new file mode 100644
index 0000000..616971d
--- /dev/null
+++ b/tests/test_hull.scad
@@ -0,0 +1,100 @@
+include <../std.scad>
+include <../hull.scad>
+
+module test_hull() {
+    assert_equal(hull([[3,4],[5,5]]), [0,1]);
+    assert_equal(hull([[3,4,1],[5,5,3]]), [0,1]);
+
+    test_collinear_2d = let(u = unit([5,3]))    [ for(i = [9,2,3,4,5,7,12,15,13]) i * u ];
+    assert_equal(hull(test_collinear_2d), [7,1]);
+    test_collinear_3d = let(u = unit([5,3,2]))    [ for(i = [9,2,3,4,5,7,12,15,13]) i * u ];
+    assert_equal(hull(test_collinear_3d), [7,1]);
+
+    /*    // produces some extra points along edges
+    test_square_2d = [for(x=[1:5], y=[2:6]) [x,y]];
+    echo(test_square_2d);
+    move_copies(test_square_2d) circle(r=.1,$fn=16);
+    color("red")move_copies(select(test_square_2d,hull(test_square_2d))) circle(r=.1,$fn=16);
+    */
+
+    /*  // also produces extra points along edges
+    test_square_2d = rot(22,p=[for(x=[1:5], y=[2:6]) [x,y]]);
+    echo(test_square_2d);
+    move_copies(test_square_2d) circle(r=.1,$fn=16);
+    color("red")move_copies(select(test_square_2d,hull(test_square_2d))) circle(r=.1,$fn=16);
+    */
+
+    rand10_2d = [[1.55356, -1.98965], [4.23157, -0.947788], [-4.06193, -1.55463],
+                 [1.23889, -3.73133], [-1.02637, -4.0155], [4.26806, -4.61909],
+                 [3.59556, -3.1574], [-2.77776, -4.21857], [-3.66253,-4.34458], [1.82324, 0.102025]];
+    assert_equal(sort(hull(rand10_2d)), [1,2,5,8,9]);
+
+    rand75_2d = [[-3.14743, -3.28139], [0.15343, -0.370249], [0.082565, 3.95939], [-2.56925, -3.16262], [-1.59463, 4.20893],
+                 [-4.90744, -1.21374], [-1.0819, -1.93703], [-3.72723, -3.0744], [-3.34339, 1.53535], [3.15803, -0.307388], [4.23289,
+                 4.46259], [1.73624, 1.38918], [3.72087, -1.55028], [1.2604, 2.30502], [-0.966431, 1.673], [-3.26866, -0.531443], [1.52605,
+                 0.991804], [-1.26305, 1.0737], [-4.31943, 4.11932], [0.488101, 0.0425981], [1.0233, -0.723037], [-4.73406, 2.14568],
+                 [-4.75915, 3.83262], [4.90999, -2.76668], [1.91971, -3.8604], [4.38594, -0.761767], [-0.352984, 1.55291], [2.02714,
+                 -0.340099], [1.76052, 2.09196], [-1.27485, -4.39477], [4.36364, 3.84964], [0.593612, -4.00028], [3.06833, -3.67117],
+                 [4.26834, -4.21213], [4.60226, -0.120432], [-2.45646, 2.60327], [-4.79461, 3.83724], [-3.29755, 0.760159], [0.218423,
+                 4.1687], [-0.115829, -2.06242], [-3.96188, 3.21568], [4.3018, -2.5299], [-4.41694, 4.75173], [-3.8393, 2.82212], [-1.14268,
+                 1.80751], [2.05805, 1.68593], [-3.0159, -2.91139], [-1.44828, -1.93564], [-0.265887, 0.519893], [-0.457361, -0.610096],
+                 [-0.426359, -2.37315], [-3.1018, 2.31141], [0.179141, -3.56242], [-0.491786, 0.813055], [-3.28502, -1.18933], [0.0914813,
+                 2.16122], [4.5777, 4.83972], [-1.07096, 2.74992], [-0.698689, 3.9032], [-1.21809, -1.54434], [3.14457, 4.92302], [-4.63176,
+                 2.81952], [4.84414, 4.63699], [2.4259, -0.747268], [-1.52088, -4.58305], [1.6961, -3.73678], [-0.483003, -3.67283],
+                 [-3.72746, -0.284265], [2.07629, 1.99902], [-3.12698, -0.96353], [4.02254, 3.41521], [-0.963391, -3.2143], [0.315255,
+                 0.593049], [1.57006, 1.80436], [4.60957, -2.86325]];
+    assert_equal(sort(hull(rand75_2d)),[5,7,23,33,36,42,56,60,62,64]);
+
+    rand10_2d_rot = rot([22,44,12], p=path3d(rand10_2d));
+    assert_equal(sort(hull(rand10_2d_rot)), [1,2,5,8,9]);
+
+    rand75_2d_rot = rot([122,-44,32], p=path3d(rand75_2d));
+    assert_equal(sort(hull(rand75_2d_rot)), [5,7,23,33,36,42,56,60,62,64]);
+
+    testpoints_on_sphere = [ for(p = 
+        [
+            [1,PHI,0], [-1,PHI,0], [1,-PHI,0], [-1,-PHI,0],
+            [0,1,PHI], [0,-1,PHI], [0,1,-PHI], [0,-1,-PHI],
+            [PHI,0,1], [-PHI,0,1], [PHI,0,-1], [-PHI,0,-1]
+        ])
+        unit(p)
+    ];
+    assert_equal(hull(testpoints_on_sphere),  [[8, 4, 0], [0, 4, 1], [4, 8, 5], [8, 2, 5], [2, 3, 5], [0, 1, 6], [3, 2, 7], [1, 4, 9], [4, 5, 9],
+                 [5, 3, 9], [8, 0, 10], [2, 8, 10], [0, 6, 10], [6, 7, 10], [7, 2, 10], [6, 1, 11], [3, 7, 11], [7, 6, 11], [1, 9, 11], [9, 3, 11]]);
+
+    rand10_3d = [[14.0893, -15.2751, 21.0843], [-14.1564, 17.5751, 3.32094], [17.4966, 12.1717, 18.0607], [24.5489, 9.64591, 10.4738], [-12.0233, -24.4368, 13.1614],
+                 [6.24019, -18.4135, 24.9554], [11.9438, -15.9724, -22.6454], [11.6147, 7.56059, 7.5667], [-19.7491, 9.42769, 15.3419], [-10.3726, 16.3559, 3.38503]];
+    assert_equal(hull(rand10_3d),[[3, 6, 0], [1, 3, 2], [3, 0, 2], [6, 1, 4], [0, 6, 5], [6, 4, 5], [2, 0, 5], [1, 2, 8], [2, 5, 8], [4, 1, 8], [5, 4, 8], [6, 3, 9], [3, 1, 9], [1, 6, 9]]);
+
+    rand25_3d = [[-20.5261, 14.5058, -11.6349], [16.4625, 20.1316, 12.9816], [-14.0268, 5.58802, 17.686], [-5.47944, 16.2501,
+                 5.3086], [20.2168, -11.8466, 12.4598], [14.4633, -15.1479, 4.82151], [12.7897, 5.25704, 19.6205], [11.2456,
+                 18.2794, -3.47074], [-1.87665, 22.9852, 1.99367], [-15.6052, -2.11009, 14.0096], [-10.7389, -14.569,
+                 5.6121], [24.5965, 17.9039, 20.8313], [-13.7054, 13.3362, 1.50374], [10.1111, -23.1494, 19.9305], [14.154,
+                 19.6682, -0.170182], [-22.6438, 22.7429, -0.776773], [-9.75056, 17.8896, -8.04152], [23.1746, 20.5475,
+                 22.6957], [-10.5356, -4.32407, -7.0911], [2.20779, -8.30749, 6.87185], [23.2643, 2.64462, -19.0087],
+                 [24.4055, 24.4504, 23.4777], [-3.84086, -6.98473, -10.2889], [0.178043, -16.07, 16.8081], [-8.86482,
+                 -12.8256, 14.7418], [11.1759, -11.5614, -11.643], [7.16751, 13.9344, -19.1675], [2.26602, -10.5374,
+                 0.125718], [-13.9053, 11.1143, -21.9289], [24.9018, -23.5307, -21.4684], [-13.6609, -19.6495, -8.91583],
+                 [-16.5393, -22.4105, -6.91617], [-4.11378, -3.14362, -5.6881], [7.50883, -17.5284, -0.0615319], [-7.41739,
+                 0.0721313, -7.47111], [22.6975, -7.99655, 14.0555], [-13.3644, 9.26993, 20.858], [-13.6889, 16.7462,
+                 -14.5836], [16.5137, 3.90703, -5.49396], [-6.75614, -11.1444, -24.5309], [22.9868, 10.0028, 12.2866],
+                 [-4.81079, -0.967785, -10.4726], [-0.949023, 23.1441, -2.08208], [16.1256, -8.2295, -24.0113], [6.45274,
+                 -7.21416, 23.1409], [22.8274, 1.07038, 19.1756], [-10.6256, -10.0112, -6.12274], [6.29254, -7.81875,
+                 -24.4037], [22.8538, 8.78163, -6.82567], [-1.96142, 19.1728, -1.726]];
+    assert_equal(hull(rand25_3d),[[21, 29, 11], [29, 21, 20], [21, 14, 20], [20, 14, 26], [15, 0, 28], [13, 29, 31], [0, 15,
+                                 31], [15, 9, 31], [9, 24, 31], [24, 13, 31], [28, 0, 31], [11, 29, 35], [29, 13, 35], [15,
+                                 21, 36], [9, 15, 36], [24, 9, 36], [13, 24, 36], [15, 28, 37], [28, 26, 37], [28, 31, 39],
+                                 [31, 29, 39], [14, 21, 42], [21, 15, 42], [26, 14, 42], [15, 37, 42], [37, 26, 42], [29, 20,
+                                 43], [39, 29, 43], [20, 26, 43], [26, 28, 43], [21, 13, 44], [13, 36, 44], [36, 21, 44],
+                                 [21, 11, 45], [11, 35, 45], [13, 21, 45], [35, 13, 45], [28, 39, 47], [39, 43, 47], [43, 28, 47]]);
+
+    /*  // Inconsistently treats coplanar faces: sometimes face center vertex is included in output, sometimes not
+    test_cube_3d = [for(x=[1:3], y=[1:3], z=[1:3]) [x,y,z]];
+    assert_equal(hull(test_cube_3d),  [[3, 2, 0], [2, 3, 4], [26, 2, 5], [2, 4, 5], [4, 3, 6], [5, 4, 6], [5, 6, 7], [6, 26, 7], [26, 5, 8],
+                                       [5, 7, 8], [7, 26, 8], [0, 2, 9], [3, 0, 9], [6, 3, 9], [9, 2, 10], [2, 26, 11], [10, 2, 11], [6, 9, 12],
+                                       [26, 6, 15], [6, 12, 15], [9, 10, 18], [10, 11, 18], [12, 9, 18], [15, 12, 18], [26, 18, 19], [18, 11, 19],
+                                       [11, 26, 20], [26, 19, 20], [19, 11, 20], [15, 18, 21], [18, 26, 21], [26, 15, 24], [15, 21, 24], [21, 26, 24]]);
+                                       echo(len=len(hull(test_cube_3d)));
+    */                                   
+}
+test_hull();