update doc

README.md

@@ -1,8 +1,8 @@
-# dotSCAD 1.1
+# dotSCAD 1.2
 
-> Helpful modules and functions when playing OpenSCAD.
+> Helpful modules and functions when playing OpenSCAD. Based on OpenSCAD 2015.03.
 
-![dotSCAD](GoldenTaiwan.JPG)
+![dotSCAD](WhirlingTaiwan.JPG)
 
 [![license/LGPL](https://img.shields.io/badge/license-LGPL-blue.svg)](https://github.com/JustinSDK/lib-openscad/blob/master/LICENSE)
 
@@ -49,6 +49,7 @@ Too many dependencies? Because OpenSCAD doesn't provide namespace management, I
 	- [hull_polyline3d](https://openhome.cc/eGossip/OpenSCAD/lib-hull_polyline3d.html)
 	- [function_grapher](https://openhome.cc/eGossip/OpenSCAD/lib-function_grapher.html)
 	- [polysections](https://openhome.cc/eGossip/OpenSCAD/lib-polysections.html)
+	- [starburst](https://openhome.cc/eGossip/OpenSCAD/lib-starburst.html)
 	
 - Transformation
     - [along_with](https://openhome.cc/eGossip/OpenSCAD/lib-along_with.html)
@@ -63,6 +64,8 @@ Too many dependencies? Because OpenSCAD doesn't provide namespace management, I
 	- [parse_number](https://openhome.cc/eGossip/OpenSCAD/lib-parse_number.html)
 	- [cross_sections](https://openhome.cc/eGossip/OpenSCAD/lib-cross_sections.html)
 	- [paths2sections](https://openhome.cc/eGossip/OpenSCAD/lib-paths2sections.html)
+	- [path_scaling_sections](https://openhome.cc/eGossip/OpenSCAD/lib-path_scaling_sections.html)
+	- [bijection_offset](https://openhome.cc/eGossip/OpenSCAD/lib-bijection_offset.html)
 	
 - Path
     - [arc_path](https://openhome.cc/eGossip/OpenSCAD/lib-arc_path.html)
@@ -74,6 +77,7 @@ Too many dependencies? Because OpenSCAD doesn't provide namespace management, I
     - [golden_spiral](https://openhome.cc/eGossip/OpenSCAD/lib-golden_spiral.html)
     - [archimedean_spiral](https://openhome.cc/eGossip/OpenSCAD/lib-archimedean_spiral.html)
     - [sphere_spiral](https://openhome.cc/eGossip/OpenSCAD/lib-sphere_spiral.html)
+	- [torus_knot](https://openhome.cc/eGossip/OpenSCAD/lib-torus_knot.html)
 
 - Extrusion
     - [box_extrude](https://openhome.cc/eGossip/OpenSCAD/lib-box_extrude.html)
@@ -104,7 +108,6 @@ Too many dependencies? Because OpenSCAD doesn't provide namespace management, I
 	- [sphere_spiral_extrude](https://openhome.cc/eGossip/OpenSCAD/lib-sphere_spiral_extrude.html)
 
 - Matrix
-	- [m_multiply](https://openhome.cc/eGossip/OpenSCAD/lib-m_multiply.html)
 	- [m_cumulate](https://openhome.cc/eGossip/OpenSCAD/lib-m_cumulate.html)	
 	- [m_translation](https://openhome.cc/eGossip/OpenSCAD/lib-m_translation.html)
 	- [m_rotation](https://openhome.cc/eGossip/OpenSCAD/lib-m_rotation.html)

RELEASE.md

@@ -1,3 +1,28 @@
+> Version numbers are based on [Semantic Versioning](https://semver.org/).
+
+# v1.2.1
+- Bugfixes
+   - Fixed CCW faces when using `path_scaling_sections`.
+
+# v1.2
+- New modules and functions:
+  - [starburst](https://openhome.cc/eGossip/OpenSCAD/lib-starburst.html)
+  - [torus_knot](https://openhome.cc/eGossip/OpenSCAD/lib-torus_knot.html)
+  - [bijection_offset](https://openhome.cc/eGossip/OpenSCAD/lib-bijection_offset.html)
+  - [path_scaling_sections](https://openhome.cc/eGossip/OpenSCAD/lib-path_scaling_sections.html)
+
+- Others
+  - Avoid warnings when using newer versions of OpenSCAD after 2015.03.
+
+# v1.1.1
+- Bugfixes
+  - `m_rotation` returns an identity matrix if `a` is 0.
+  - The `path_pts` parameter of `path_extrude` accepts two or three points. 
+  - The `points` parameter of `along_with` accepts two or three points. 
+
+- Others
+  -  OpenSCAD has built-in matrix multiplication so `m_multiply` is not necessary.
+
 # v1.1
 - New matrix functions:
   - [m_multiply](https://openhome.cc/eGossip/OpenSCAD/lib-m_multiply.html)

docs/lib-arc_path.md

@@ -15,8 +15,8 @@ Creates an arc path. You can pass a 2 element vector to define the central angle
     include <hull_polyline2d.scad>;
     
     $fn = 24;
-    points = arc_path(radius = 20, angle = [45, 290], width = 2);
-    hull_polyline2d(points);
+    points = arc_path(radius = 20, angle = [45, 290]);
+    hull_polyline2d(points, width = 2);
 
 ![arc_path](images/lib-arc_path-1.JPG)
 
@@ -24,8 +24,8 @@ Creates an arc path. You can pass a 2 element vector to define the central angle
     include <hull_polyline2d.scad>;
     
     $fn = 24;
-    points = arc_path(radius = 20, angle = 135, width = 2);
-    hull_polyline2d(points);
+    points = arc_path(radius = 20, angle = 135);
+    hull_polyline2d(points, width = 2);
 
 ![arc_path](images/lib-arc_path-2.JPG)
 

docs/lib-bijection_offset.md

@@ -0,0 +1,67 @@
+# bijection_offset
+
+Move 2D outlines outward or inward by a given amount. Each point of the offsetted shape is paired with exactly one point of the original shape.
+
+**Since:** 1.2.
+
+## Parameters
+
+- `pts` : Points of a shape.
+- `d` : Amount to offset the shape. When negative, the shape is offset inwards. 
+
+## Examples
+
+	include <bijection_offset.scad>;
+
+	shape = [
+		[15, 0],
+		[15, 30],
+		[0, 20],
+		[-15, 40],
+		[-15, 0]
+	];
+
+	color("red") polygon(bijection_offset(shape, 3));
+	color("orange") polygon(bijection_offset(shape, 2));
+	color("yellow") polygon(bijection_offset(shape, 1));
+	color("green") polygon(shape);
+	color("blue") polygon(bijection_offset(shape, -1));
+	color("indigo") polygon(bijection_offset(shape, -2));
+	color("purple") polygon(bijection_offset(shape, -3));
+
+![bijection_offset](images/lib-bijection_offset-1.JPG)
+
+	include <bijection_offset.scad>;
+	include <rotate_p.scad>;
+	include <polysections.scad>;
+	include <path_extrude.scad>;
+	include <bezier_curve.scad>;
+
+	shape = [
+		[5, 0],
+		[3, 9],
+		[0, 10],    
+		[-5, 0]
+	];
+	offsetted = bijection_offset(shape, 1);
+
+	offsetted2 = bijection_offset(shape, 2);
+	offsetted3 = bijection_offset(shape, 3);
+
+	t_step = 0.05;
+
+	p0 = [0, 0, 0];
+	p1 = [40, 60, 35];
+	p2 = [-50, 70, 0];
+	p3 = [20, 150, -35];
+	p4 = [30, 50, -3];
+
+	path_pts = bezier_curve(t_step, 
+		[p0, p1, p2, p3, p4]
+	);
+
+	path_extrude(concat(offsetted, shape), path_pts, "HOLLOW");
+	path_extrude(concat(offsetted3, offsetted2), path_pts, "HOLLOW");
+
+![bijection_offset](images/lib-bijection_offset-2.JPG)
+

docs/lib-m_mirror.md

@@ -1,23 +0,0 @@
-# m_mirror
-
-Generate a 4x4 transformation matrix which can pass into `multmatrix` to mirror the child element on a plane through the origin. 
-
-**Since:** 1.1
-
-## Parameters
-
-- `v` : The normal vector of a plane intersecting the origin through which to mirror the object.
-
-## Examples
-
-	include <m_mirror.scad>;
-
-	rotate([0, 0, 10]) 
-		cube([3, 2, 1]);
-		
-	multmatrix(m_mirror([1, 1, 0]))
-		rotate([0, 0, 10]) 
-			cube([3, 2, 1]);
-
-![m_mirror](images/lib-m_mirror-1.JPG)
-

docs/lib-path_extrude.md

@@ -13,7 +13,7 @@ When using this module, you should use points to represent the 2D shape. If your
 - `triangles` : `"SOLID"` (default), `"HOLLOW"` or user-defined indexes. See example below.
 - `twist` : The number of degrees of through which the shape is extruded.
 - `scale` : Scales the 2D shape by this value over the length of the extrusion. Scale can be a scalar or a vector.
-- `closed` : If the first point and the last point of `path_pts` has the same coordinate, setting `closed` to `true` will connect them automatically.
+- `closed` : If the first point and the last point of `path_pts` has the same coordinate, setting `closed` to `true` will connect them automatically. You might have to set `twist` for connecting naturally. 
 
 ## Examples
 

docs/lib-path_scaling_sections.md

@@ -0,0 +1,133 @@
+# path_scaling_sections
+
+Given an edge path with the first point at the outline of a shape. This function uses the path to calculate scaling factors and returns all scaled sections in the reversed order of the edge path. Combined with the `polysections` module, you can create an extrusion with the path as an edge. 
+
+In order to control scaling factors easily, I suggest using `[x, 0, 0]` as the first point and keeping y = 0 while building the edge path.
+
+You can use any point as the first point of the edge path. Just remember that your edge path radiates from the origin.
+
+**Since:** 1.2.
+
+## Parameters
+
+- `shape_pts` : A list of points represent a shape.
+- `edge_path` : A list of points represent the edge path.
+
+## Examples
+
+	include <hull_polyline3d.scad>;
+	include <shape_taiwan.scad>;
+	include <path_scaling_sections.scad>;
+	include <m_scaling.scad>;
+	include <polysections.scad>;
+
+	taiwan = shape_taiwan(100);
+	fst_pt = [13, 0, 0];
+
+	edge_path = [
+		fst_pt,
+		fst_pt + [0, 0, 10],
+		fst_pt + [10, 0, 20],
+		fst_pt + [8, 0, 30],
+		fst_pt + [12, 0, 40],
+		fst_pt + [0, 0, 50],
+		fst_pt + [0, 0, 60]
+	];
+
+	#hull_polyline3d(edge_path);
+	polysections(path_scaling_sections(taiwan, edge_path));
+
+![path_scaling_sections](images/lib-path_scaling_sections-1.JPG)
+
+	include <hull_polyline3d.scad>;
+	include <shape_taiwan.scad>;
+	include <path_scaling_sections.scad>;
+	include <m_scaling.scad>;
+	include <polysections.scad>;
+	include <bezier_curve.scad>;
+
+
+	taiwan = shape_taiwan(100);
+	fst_pt = [13, 0, 0];
+
+	edge_path = bezier_curve(0.05, [
+		fst_pt,
+		fst_pt + [0, 0, 10],
+		fst_pt + [10, 0, 20],
+		fst_pt + [8, 0, 30],
+		fst_pt + [12, 0, 40],
+		fst_pt + [0, 0, 50],
+		fst_pt + [0, 0, 60]
+	]);
+
+	#hull_polyline3d(edge_path);
+	polysections(path_scaling_sections(taiwan, edge_path));
+
+![path_scaling_sections](images/lib-path_scaling_sections-2.JPG)
+
+	include <shape_taiwan.scad>;
+	include <path_scaling_sections.scad>;
+	include <m_scaling.scad>;
+	include <polysections.scad>;
+	include <bezier_curve.scad>;
+	include <rotate_p.scad>;
+
+	taiwan = shape_taiwan(100);
+	fst_pt = [13, 0, 0];
+
+	edge_path = bezier_curve(0.05, [
+		fst_pt,
+		fst_pt + [0, 0, 10],
+		fst_pt + [10, 0, 20],
+		fst_pt + [8, 0, 30],
+		fst_pt + [12, 0, 40],
+		fst_pt + [0, 0, 50],
+		fst_pt + [0, 0, 60]
+	]);
+
+	leng = len(edge_path);
+	twist = -90;
+	twist_step = twist / leng;
+	sections = path_scaling_sections(taiwan, edge_path);
+
+	rotated_sections = [
+		for(i = [0:leng - 1]) 
+		[
+			for(p = sections[i]) 
+				rotate_p(p, twist_step * i)        
+		]
+	];
+
+	polysections(rotated_sections);
+
+![path_scaling_sections](images/lib-path_scaling_sections-3.JPG)	
+
+	include <hull_polyline3d.scad>;
+	include <shape_taiwan.scad>;
+	include <path_scaling_sections.scad>;
+	include <m_scaling.scad>;
+	include <polysections.scad>;
+    include <rotate_p.scad>;
+
+	taiwan = shape_taiwan(100);
+
+    /* 
+	    You can use any point as the first point of the edge path.
+		Just remember that your edge path radiates from the origin.
+	*/
+	fst_pt = [taiwan[0][0], taiwan[0][1], 0];//[13, 0, 0];
+    a = atan2(fst_pt[1], fst_pt[0]);
+	edge_path = [
+		fst_pt,
+		fst_pt + rotate_p([0, 0, 10], a),
+		fst_pt + rotate_p([10, 0, 20], a),
+		fst_pt + rotate_p([8, 0, 30], a),
+		fst_pt + rotate_p([10, 0, 40], a),
+		fst_pt + rotate_p([0, 0, 50], a),
+		fst_pt + rotate_p([0, 0, 60], a)
+	];
+
+	#hull_polyline3d(edge_path);
+	polysections(path_scaling_sections(taiwan, edge_path));
+
+![path_scaling_sections](images/lib-path_scaling_sections-4.JPG)

docs/lib-shape_cyclicpolygon.md

@@ -6,6 +6,7 @@ Returns shape points of a regular cyclic polygon. They can be used with xxx_extr
 
 - `sides` : The radius of the circle.
 - `circle_r` : The radius of the circumcircle.
+- `corner_r` : The radius of the circle at a corner.
 - `$fa`, `$fs`, `$fn` : Check [the circle module](https://en.wikibooks.org/wiki/OpenSCAD_User_Manual/Using_the_2D_Subsystem#circle) for more details.
 
 ## Examples

docs/lib-shape_starburst.md

@@ -6,7 +6,7 @@ Returns shape points of a star. They can be used with xxx_extrude modules of dot
 
 - `r1` : The outer radius of the starburst. 
 - `r2` : The inner radius of the starburst.
-- `n`  : The number of vertices. 
+- `n`  : The burst number. 
 
 
 ## Examples

docs/lib-starburst.md

@@ -0,0 +1,24 @@
+# starburst 
+
+A 3D version of `shape_starburst`. 
+
+**Since:** 1.2.
+
+## Parameters
+
+- `r1` : The outer radius of the starburst. 
+- `r2` : The inner radius of the starburst.
+- `n`  : The number of vertices. 
+- `height` : The height of the starburst.
+
+## Examples
+
+    include <starburst.scad>;
+
+	starburst(10, 5, 5, 5);
+	translate([20, 0, 0]) starburst(10, 5, 6, 5);
+	translate([40, 0, 0]) starburst(10, 5, 12, 10);
+	translate([60, 0, 0]) starburst(10, 5, 4, 3);
+
+![starburst](images/lib-starburst-1.JPG)
+

docs/lib-torus_knot.md

@@ -0,0 +1,40 @@
+# torus_knot 
+
+Generate a path of [The (p,q)-torus knot](https://en.wikipedia.org/wiki/Torus_knot).
+
+**Since:** 1.2.
+
+![torus_knot](images/lib-torus_knot-1.JPG)
+
+## Parameters
+
+- `p` : The p parameter of The (p,q)-torus knot. 
+- `q` : The q parameter of The (p,q)-torus knot. 
+- `phi_step`  : The amount when increasing phi. 
+
+## Examples
+
+	include <shape_pentagram.scad>;
+	include <rotate_p.scad>;
+	include <polysections.scad>;
+	include <path_extrude.scad>;
+	include <torus_knot.scad>;
+
+	p = 2;
+	q = 3;
+	phi_step = 0.05;
+	star_radius = 0.5;
+
+	pts = torus_knot(p, q, phi_step);
+
+	shape_pentagram_pts = shape_pentagram(star_radius);
+
+	path_extrude(
+		shape_pentagram_pts, 
+		concat(pts, [pts[0]]), 
+		closed = true,
+		twist = 188
+	);
+
+![torus_knot](images/lib-torus_knot-2.JPG)
+

src/__private__/__half_trapezium.scad

@@ -44,9 +44,9 @@ function __br_corner(frags, b_ang, l1, l2, h, round_r) =
 
 function __half_trapezium(length, h, round_r) =
     let(
-        is_vt = __is_vector(length),
-        l1 = is_vt ? length[0] : length,
-        l2 = is_vt ? length[1] : length,
+        is_flt = __is_float(length),
+        l1 = is_flt ? length : length[0],
+        l2 = is_flt ? length : length[1],
         frags = __frags(round_r),
         b_ang = atan2(h, l1 - l2),
         br_corner = __br_corner(frags, b_ang, l1, l2, h, round_r),

src/__private__/__m_multiply.scad

@@ -1,13 +0,0 @@
-function __m_multiply(ma, mb) = 
-    let(
-        c1 = [mb[0][0], mb[1][0], mb[2][0], mb[3][0]],
-        c2 = [mb[0][1], mb[1][1], mb[2][1], mb[3][1]],
-        c3 = [mb[0][2], mb[1][2], mb[2][2], mb[3][2]],
-        c4 = [mb[0][3], mb[1][3], mb[2][3], mb[3][3]]
-    )
-    [
-        [ma[0] * c1, ma[0] * c2, ma[0] * c3, ma[0] * c4],
-        [ma[1] * c1, ma[1] * c2, ma[1] * c3, ma[1] * c4],
-        [ma[2] * c1, ma[2] * c2, ma[2] * c3, ma[2] * c4],
-        [ma[3] * c1, ma[3] * c2, ma[3] * c3, ma[3] * c4]
-    ];

src/__private__/__shape_arc.scad

@@ -6,7 +6,7 @@ function __shape_arc(radius, angle, width, width_mode = "LINE_CROSS") =
         frags = __frags(radius),
         a_step = 360 / frags,
         half_a_step = a_step / 2,
-        angles = __is_vector(angle) ? angle : [0, angle],
+        angles = __is_float(angle) ? [0, angle] : angle,
         m = floor(angles[0] / a_step) + 1,
         n = floor(angles[1] / a_step),
         r_outer = radius + w_offset[0],

src/__private__/__shape_pie.scad

@@ -3,7 +3,7 @@ function __shape_pie(radius, angle) =
         frags = __frags(radius),
         a_step = 360 / frags,
         leng = radius * cos(a_step / 2),
-        angles = __is_vector(angle) ? angle : [0:angle],
+        angles = __is_float(angle) ? [0:angle] : angle,
         m = floor(angles[0] / a_step) + 1,
         n = floor(angles[1] / a_step),
         edge_r_begin = leng / cos((m - 0.5) * a_step - angles[0]),

src/__private__/__to_ang_vect.scad

@@ -0,0 +1,7 @@
+function __to_3_elems_ang_vect(a) =
+     let(leng = len(a))
+     leng == 3 ? a : (
+         leng == 2 ? [a[0], a[1], 0] :  [a[0], 0, 0]
+     );
+
+function __to_ang_vect(a) = __is_float(a) ? [0, 0, a] : __to_3_elems_ang_vect(a);

src/along_with.scad

@@ -7,11 +7,10 @@
 * @see https://openhome.cc/eGossip/OpenSCAD/lib-along_with.html
 *
 **/ 
-
+ 
 include <__private__/__angy_angz.scad>;
-include <__private__/__is_vector.scad>;
+include <__private__/__is_float.scad>;
 include <__private__/__to3d.scad>;
-include <__private__/__m_multiply.scad>;
 
 // Becuase of improving the performance, this module requires m_rotation.scad which doesn't require in dotSCAD 1.0. 
 // For backward compatibility, I directly include m_rotation here.
@@ -26,15 +25,23 @@ module along_with(points, angles, twist = 0, scale = 1.0) {
         let(s =  (scale - 1) / leng_points_minus_one)
         [s, s, s];
 
-    scale_step_vt = __is_vector(scale) ? 
+    scale_step_vt = __is_float(scale) ? 
+        scale_step() :
         [
             (scale[0] - 1) / leng_points_minus_one, 
             (scale[1] - 1) / leng_points_minus_one,
             scale[2] == undef ? 0 : (scale[2] - 1) / leng_points_minus_one
-        ] : scale_step(); 
+        ]; 
 
     // get rotation matrice for sections
 
+    identity_matrix = [
+        [1, 0, 0, 0],
+        [0, 1, 0, 0],
+        [0, 0, 1, 0],
+        [0, 0, 0, 1]
+    ];    
+
     function local_ang_vects(j) = 
         j == 0 ? [] : local_ang_vects_sub(j);
     
@@ -53,12 +60,16 @@ module along_with(points, angles, twist = 0, scale = 1.0) {
             leng_rot_matrice_minus_one = leng_rot_matrice - 1,
             leng_rot_matrice_minus_two = leng_rot_matrice - 2
         )
-        i == leng_rot_matrice - 2 ? 
+        leng_rot_matrice == 0 ? [identity_matrix] : (
+            leng_rot_matrice == 1 ? [rot_matrice[0], identity_matrix] : (
+                i == leng_rot_matrice_minus_two ? 
                [
                    rot_matrice[leng_rot_matrice_minus_one], 
-                   __m_multiply(rot_matrice[leng_rot_matrice_minus_two], rot_matrice[leng_rot_matrice_minus_one])
+                   rot_matrice[leng_rot_matrice_minus_two] * rot_matrice[leng_rot_matrice_minus_one]
                ] 
-               : cumulated_rot_matrice_sub(i, rot_matrice);
+               : cumulated_rot_matrice_sub(i, rot_matrice)
+            )
+        );
 
     function cumulated_rot_matrice_sub(i, rot_matrice) = 
         let(
@@ -66,7 +77,7 @@ module along_with(points, angles, twist = 0, scale = 1.0) {
             curr_matrix = rot_matrice[i],
             prev_matrix = matrice[len(matrice) - 1]
         )
-        concat(matrice, [__m_multiply(curr_matrix, prev_matrix)]);
+        concat(matrice, [curr_matrix * prev_matrix]);
 
     // align modules
 

src/arc.scad

@@ -9,7 +9,7 @@
 **/ 
 
 include <__private__/__frags.scad>;
-include <__private__/__is_vector.scad>;
+include <__private__/__is_float.scad>;
 include <__private__/__ra_to_xy.scad>;
 include <__private__/__edge_r.scad>;
 include <__private__/__shape_arc.scad>;

src/arc_path.scad

@@ -9,7 +9,7 @@
 **/ 
 
 include <__private__/__frags.scad>;
-include <__private__/__is_vector.scad>;
+include <__private__/__is_float.scad>;
 include <__private__/__ra_to_xy.scad>;
 include <__private__/__edge_r.scad>;
 
@@ -17,7 +17,7 @@ function arc_path(radius, angle) =
     let(
         frags = __frags(radius),
         a_step = 360 / frags,
-        angles = __is_vector(angle) ? angle : [0, angle],
+        angles = __is_float(angle) ? [0, angle] : angle,
         m = floor(angles[0] / a_step) + 1,
         n = floor(angles[1] / a_step),
         points = concat([__ra_to_xy(__edge_r_begin(radius, angles[0], a_step, m), angles[0])],

src/bijection_offset.scad

@@ -0,0 +1,86 @@
+/**
+* bijection_offset.scad
+*
+* @copyright Justin Lin, 2019
+* @license https://opensource.org/licenses/lgpl-3.0.html
+*
+* @see https://openhome.cc/eGossip/OpenSCAD/lib-bijection_offset.html
+*
+**/
+
+function _bijection_edges_from(pts) = 
+    let(leng = len(pts))
+    concat(
+        [for(i = [0:leng - 2]) [pts[i], pts[i + 1]]], 
+        [[pts[len(pts) - 1], pts[0]]]
+    );
+    
+function _bijection_inward_edge_normal(edge) =  
+    let(
+        pt1 = edge[0],
+        pt2 = edge[1],
+        dx = pt2[0] - pt1[0],
+        dy = pt2[1] - pt1[1],
+        edge_leng = norm([dx, dy])
+    )
+    [-dy / edge_leng, dx / edge_leng];
+
+function _bijection_outward_edge_normal(edge) = -1 * _bijection_inward_edge_normal(edge);
+
+function _bijection_offset_edge(edge, dx, dy) = 
+    let(
+        pt1 = edge[0],
+        pt2 = edge[1],
+        dxy = [dx, dy]
+    )
+    [pt1 + dxy, pt2 + dxy];
+    
+function _bijection__bijection_offset_edges(edges, d) = 
+    [ 
+        for(edge = edges)
+        let(
+            ow_normal = _bijection_outward_edge_normal(edge),
+            dx = ow_normal[0] * d,
+            dy = ow_normal[1] * d
+        )
+        _bijection_offset_edge(edge, dx, dy)
+    ];
+    
+
+function _bijection__bijection__bijection_offset_edges_intersection(edge1, edge2) = 
+    let(
+        den = (edge2[1][1] - edge2[0][1]) * (edge1[1][0] - edge1[0][0]) - (edge2[1][0] - edge2[0][0]) * (edge1[1][1] - edge1[0][1])
+    )
+    // when den is 0, they are parallel or conincident edges
+    den == 0 ? [] : _bijection_offset__bijection__bijection__bijection_offset_edges_intersection_sub(edge1, edge2, den);
+
+function _bijection_offset__bijection__bijection__bijection_offset_edges_intersection_sub(edge1, edge2, den) = 
+    let(
+        ua = ((edge2[1][0] - edge2[0][0]) * (edge1[0][1] - edge2[0][1]) - (edge2[1][1] - edge2[0][1]) * (edge1[0][0] - edge2[0][0])) / den
+    ) 
+    [
+        edge1[0][0] + ua * (edge1[1][0] - edge1[0][0]),
+        edge1[0][1] + ua * (edge1[1][1] - edge1[0][1])
+    ];
+
+function bijection_offset(pts, d) = 
+    let(
+        es = _bijection_edges_from(pts), 
+        offset_es = _bijection__bijection_offset_edges(es, d),
+        leng = len(offset_es),
+        last_p = _bijection__bijection__bijection_offset_edges_intersection(offset_es[leng - 1], offset_es[0])
+    )
+    concat(
+        [
+            for(i = [0:leng - 2]) 
+            let(
+                this_edge = offset_es[i],
+                next_edge = offset_es[i + 1],
+                p = _bijection__bijection__bijection_offset_edges_intersection(this_edge, next_edge)
+            )
+            // p == p to avoid [nan, nan], because [nan, nan] != [nan, nan]
+            if(p != [] && p == p) p
+        ],
+        last_p != [] && last_p == last_p ? [last_p] : []
+    );
+    

src/cross_sections.scad

@@ -9,16 +9,17 @@
 **/
 
 include <__private__/__to3d.scad>;
-include <__private__/__is_vector.scad>;
+include <__private__/__is_float.scad>;
 
 function cross_sections(shape_pts, path_pts, angles, twist = 0, scale = 1.0) =
     let(
         len_path_pts_minus_one = len(path_pts) - 1,
         sh_pts = len(shape_pts[0]) == 3 ? shape_pts : [for(p = shape_pts) __to3d(p)],
         pth_pts = len(path_pts[0]) == 3 ? path_pts : [for(p = path_pts) __to3d(p)],
-        scale_step_vt = __is_vector(scale) ? 
-            [(scale[0] - 1) / len_path_pts_minus_one, (scale[1] - 1) / len_path_pts_minus_one] :
-            [(scale - 1) / len_path_pts_minus_one, (scale - 1) / len_path_pts_minus_one],
+        scale_step_vt = __is_float(scale) ? 
+            [(scale - 1) / len_path_pts_minus_one, (scale - 1) / len_path_pts_minus_one] :
+            [(scale[0] - 1) / len_path_pts_minus_one, (scale[1] - 1) / len_path_pts_minus_one]
+            ,
         scale_step_x = scale_step_vt[0],
         scale_step_y = scale_step_vt[1],
         twist_step = twist / len_path_pts_minus_one

src/crystal_ball.scad

@@ -9,10 +9,10 @@
 **/ 
 
 include <__private__/__nearest_multiple_of_4.scad>;
-include <__private__/__is_vector.scad>;
+include <__private__/__is_float.scad>;
 
 module crystal_ball(radius, theta = 360, phi = 180) {
-    phis = __is_vector(phi) ? phi : [0, phi];
+    phis = __is_float(phi) ? [0, phi] : phi;
     
     frags = __frags(radius);
 

src/helix.scad

@@ -8,14 +8,14 @@
 *
 **/ 
 
-include <__private__/__is_vector.scad>;
+include <__private__/__is_float.scad>;
 include <__private__/__frags.scad>;
 
 function helix(radius, levels, level_dist, vt_dir = "SPI_DOWN", rt_dir = "CT_CLK") = 
     let(
-        is_vt = __is_vector(radius),
-        r1 = is_vt ? radius[0] : radius,
-        r2 = is_vt ? radius[1] : radius,
+        is_flt = __is_float(radius),
+        r1 = is_flt ? radius : radius[0],
+        r2 = is_flt ? radius : radius[1],
         init_r = vt_dir == "SPI_DOWN" ? r2 : r1,
         _frags = __frags(init_r),
         h = level_dist * levels,

src/helix_extrude.scad

@@ -8,7 +8,7 @@
 *
 **/
 
-include <__private__/__is_vector.scad>;
+include <__private__/__is_float.scad>;
 include <__private__/__frags.scad>;
 
 module helix_extrude(shape_pts, radius, levels, level_dist, 
@@ -22,9 +22,9 @@ module helix_extrude(shape_pts, radius, levels, level_dist,
                 vt[leng - 1 - i]
         ];                         
                          
-    is_vt = __is_vector(radius);
-    r1 = is_vt ? radius[0] : radius;
-    r2 = is_vt ? radius[1] : radius;
+    is_flt = __is_float(radius);
+    r1 = is_flt ? radius : radius[0];
+    r2 = is_flt ? radius : radius[1];
     
     init_r = vt_dir == "SPI_DOWN" ? r2 : r1;
 

src/m_cumulate.scad

@@ -8,12 +8,10 @@
 *
 **/
 
-include <__private__/__m_multiply.scad>;
-
 function _m_cumulate(matrice, i) = 
     i == len(matrice) - 2 ?
-        __m_multiply(matrice[i], matrice[i + 1]) :
-        __m_multiply(matrice[i], _m_cumulate(matrice, i + 1));
+        matrice[i] * matrice[i + 1] :
+        matrice[i] * _m_cumulate(matrice, i + 1);
 
 function m_cumulate(matrice) = 
     len(matrice) == 1 ? matrice[0] : _m_cumulate(matrice, 0);

src/m_multiply.scad

@@ -1,13 +0,0 @@
-/**
-* m_multiply.scad
-*
-* @copyright Justin Lin, 2019
-* @license https://opensource.org/licenses/lgpl-3.0.html
-*
-* @see https://openhome.cc/eGossip/OpenSCAD/lib-m_multiply.html
-*
-**/
-
-include <__private__/__m_multiply.scad>;
-
-function m_multiply(ma, mb) = __m_multiply(ma, mb);

src/m_rotation.scad

@@ -8,7 +8,8 @@
 *
 **/
 
-include <__private__/__m_multiply.scad>;
+include <__private__/__is_float.scad>;
+include <__private__/__to_ang_vect.scad>;
 
 function _q_rotation(a, v) = 
     let(
@@ -68,20 +69,14 @@ function _m_zRotation(a) =
         [0, 0, 0, 1]
     ];    
 
-function _to_avect(a) =
-     len(a) == 3 ? a : (
-         len(a) == 2 ? [a[0], a[1], 0] : (
-             len(a) == 1 ? [a[0], 0, 0] : [0, 0, a]
-         ) 
-     );
-
 function _xyz_rotation(a) =
-    let(ang = _to_avect(a))
-    __m_multiply(
-        _m_zRotation(ang[2]), __m_multiply(
-            _m_yRotation(ang[1]), _m_xRotation(ang[0])
-        )
-    );
+    let(ang = __to_ang_vect(a))
+    _m_zRotation(ang[2]) * _m_yRotation(ang[1]) * _m_xRotation(ang[0]);
 
 function m_rotation(a, v) = 
-    v == undef ? _xyz_rotation(a) : _q_rotation(a, v);
+    (a == 0 || a == [0, 0, 0] || a == [0] || a == [0, 0]) ? [
+        [1, 0, 0, 0],
+        [0, 1, 0, 0],
+        [0, 0, 1, 0],
+        [0, 0, 0, 1]
+    ] : (v == undef ? _xyz_rotation(a) : _q_rotation(a, v));

src/m_scaling.scad

@@ -8,15 +8,18 @@
 *
 **/
 
-function _to_svect(s) =
-     len(s) == 3 ? s : (
-         len(s) == 2 ? [s[0], s[1], 1] : (
-             len(s) == 1 ? [s[0], 1, 1] : [s, s, s]
-         ) 
+include <__private__/__is_float.scad>;
+
+function _to_3_elems_scaling_vect(s) =
+     let(leng = len(s))
+     leng == 3 ? s : (
+         leng == 2 ? [s[0], s[1], 1] : [s[0], 1, 1]
      );
 
+function _to_scaling_vect(s) = __is_float(s) ? [s, s, s] : _to_3_elems_scaling_vect(s);
+
 function m_scaling(s) = 
-    let(v = _to_svect(s))
+    let(v = _to_scaling_vect(s))
     [
         [v[0], 0, 0, 0],
         [0, v[1], 0, 0],

src/m_shearing.scad

@@ -8,7 +8,6 @@
 *
 **/
 
-include <__private__/__m_multiply.scad>;
 include <__private__/__m_shearing.scad>;
 
 function m_shearing(sx = [0, 0], sy = [0, 0], sz = [0, 0]) = __m_shearing(sx, sy, sz);

src/m_translation.scad

@@ -8,15 +8,18 @@
 *
 **/
 
-function _to_tvect(v) =
-     len(v) == 3 ? v : (
-         len(v) == 2 ? [v[0], v[1], 0] : (
-             len(v) == 1 ? [v[0], 0, 0] : [v, 0, 0]
-         ) 
+include <__private__/__is_float.scad>;
+
+function _to_3_elems_translation_vect(v) =
+     let(leng = len(v))
+     leng == 3 ? v : (
+         leng == 2 ? [v[0], v[1], 0] : [v[0], 0, 0]
      );
 
+function _to_translation_vect(v) = __is_float(v) ? [v, 0, 0] : _to_3_elems_translation_vect(v);
+
 function m_translation(v) = 
-    let(vt = _to_tvect(v))
+    let(vt = _to_translation_vect(v))
     [
         [1, 0, 0, vt[0]],
         [0, 1, 0, vt[1]],

src/path_extrude.scad

@@ -8,10 +8,9 @@
 *
 **/
 
-include <__private__/__is_vector.scad>;
+include <__private__/__is_float.scad>;
 include <__private__/__to3d.scad>;
 include <__private__/__angy_angz.scad>;
-include <__private__/__m_multiply.scad>;
 
 // Becuase of improving the performance, this module requires m_rotation.scad which doesn't require in dotSCAD 1.0. 
 // For backward compatibility, I directly include m_rotation here.
@@ -40,12 +39,13 @@ module path_extrude(shape_pts, path_pts, triangles = "SOLID", twist = 0, scale =
         let(s =  (scale - 1) / len_path_pts_minus_one)
         [s, s, s];  
 
-    scale_step_vt = __is_vector(scale) ? 
+    scale_step_vt = __is_float(scale) ? 
+        scale_step() : 
         [
             (scale[0] - 1) / len_path_pts_minus_one, 
             (scale[1] - 1) / len_path_pts_minus_one,
             scale[2] == undef ? 0 : (scale[2] - 1) / len_path_pts_minus_one
-        ] : scale_step();   
+        ];   
 
     // get rotation matrice for sections
 
@@ -70,13 +70,24 @@ module path_extrude(shape_pts, path_pts, triangles = "SOLID", twist = 0, scale =
     leng_rot_matrice_minus_one = leng_rot_matrice - 1;
     leng_rot_matrice_minus_two= leng_rot_matrice - 2;
 
+    identity_matrix = [
+        [1, 0, 0, 0],
+        [0, 1, 0, 0],
+        [0, 0, 1, 0],
+        [0, 0, 0, 1]
+    ];
+
     function cumulated_rot_matrice(i) = 
-        i == leng_rot_matrice - 2 ? 
-               [
-                   rot_matrice[leng_rot_matrice_minus_one], 
-                   __m_multiply(rot_matrice[leng_rot_matrice_minus_two], rot_matrice[leng_rot_matrice_minus_one])
-               ] 
-               : cumulated_rot_matrice_sub(i);
+        leng_rot_matrice == 0 ? [identity_matrix] : (
+             leng_rot_matrice == 1 ? [rot_matrice[0], identity_matrix] : 
+                (
+                    i == leng_rot_matrice_minus_two ? 
+                    [
+                        rot_matrice[leng_rot_matrice_minus_one], 
+                        rot_matrice[leng_rot_matrice_minus_two] * rot_matrice[leng_rot_matrice_minus_one]
+                    ] 
+                    : cumulated_rot_matrice_sub(i))
+        );
 
     function cumulated_rot_matrice_sub(i) = 
         let(
@@ -84,7 +95,7 @@ module path_extrude(shape_pts, path_pts, triangles = "SOLID", twist = 0, scale =
             curr_matrix = rot_matrice[i],
             prev_matrix = matrice[len(matrice) - 1]
         )
-        concat(matrice, [__m_multiply(curr_matrix, prev_matrix)]);
+        concat(matrice, [curr_matrix * prev_matrix]);
 
     cumu_rot_matrice = cumulated_rot_matrice(0);
 

src/path_scaling_sections.scad

@@ -0,0 +1,30 @@
+/**
+* path_scaling_sections.scad
+*
+* @copyright Justin Lin, 2019
+* @license https://opensource.org/licenses/lgpl-3.0.html
+*
+* @see https://openhome.cc/eGossip/OpenSCAD/lib-path_scaling_sections.html
+*
+**/
+
+include <__private__/__reverse.scad>;
+
+function path_scaling_sections(shape_pts, edge_path) = 
+    let(
+        start_point = edge_path[0],
+        base_leng = norm(start_point),
+        scaling_matrice = [
+            for(p = edge_path) 
+            let(s = norm([p[0], p[1], 0]) / base_leng)
+            m_scaling([s, s, 1])
+        ]
+    )
+    __reverse([
+        for(i = [0:len(edge_path) - 1])
+        [
+            for(p = shape_pts) 
+            let(scaled_p = scaling_matrice[i] * [p[0], p[1], edge_path[i][2], 1])
+            [scaled_p[0], scaled_p[1], scaled_p[2]]
+        ]
+    ]);

src/pie.scad

@@ -9,7 +9,7 @@
 **/
 
 include <__private__/__frags.scad>;
-include <__private__/__is_vector.scad>;
+include <__private__/__is_float.scad>;
 include <__private__/__ra_to_xy.scad>;
 include <__private__/__shape_pie.scad>;
  

src/ring_extrude.scad

@@ -19,7 +19,7 @@ module ring_extrude(shape_pts, radius, angle = 360, twist = 0, scale = 1.0, tria
     } else {
         a_step = 360 / __frags(radius);
 
-        angles = __is_vector(angle) ? angle : [0, angle];
+        angles = __is_float(angle) ? [0, angle] : angle;
 
         m = floor(angles[0] / a_step) + 1;
         n = floor(angles[1] / a_step);

src/rotate_p.scad

@@ -10,7 +10,8 @@
 
 include <__private__/__to2d.scad>;
 include <__private__/__to3d.scad>;
-
+include <__private__/__is_float.scad>;
+include <__private__/__to_ang_vect.scad>;
 
 function _q_rotate_p_3d(p, a, v) = 
     let(
@@ -69,15 +70,8 @@ function _rotz(pt, a) =
 function _rotate_p_3d(point, a) =
     _rotz(_roty(_rotx(point, a[0]), a[1]), a[2]);
 
-function _to_avect(a) =
-     len(a) == 3 ? a : (
-         len(a) == 2 ? [a[0], a[1], 0] : (
-             len(a) == 1 ? [a[0], 0, 0] : [0, 0, a]
-         ) 
-     );
-
 function _rotate_p(p, a) =
-    let(angle = _to_avect(a))
+    let(angle = __to_ang_vect(a))
     len(p) == 3 ? 
         _rotate_p_3d(p, angle) :
         __to2d(

src/rounded_cube.scad

@@ -8,15 +8,15 @@
 *
 **/
 
-include <__private__/__is_vector.scad>;
+include <__private__/__is_float.scad>;
 include <__private__/__frags.scad>;
 include <__private__/__nearest_multiple_of_4.scad>;
 
 module rounded_cube(size, corner_r, center = false) {
-    is_vt = __is_vector(size);
-    x = is_vt ? size[0] : size;
-    y = is_vt ? size[1] : size;
-    z = is_vt ? size[2] : size;
+    is_flt = __is_float(size);
+    x = is_flt ? size : size[0];
+    y = is_flt ? size : size[1];
+    z = is_flt ? size : size[2];
 
     corner_frags = __nearest_multiple_of_4(__frags(corner_r));
     edge_d = corner_r * cos(180 / corner_frags);

src/rounded_cylinder.scad

@@ -8,7 +8,7 @@
 *
 **/
 
-include <__private__/__is_vector.scad>;
+include <__private__/__is_float.scad>;
 include <__private__/__frags.scad>;
 include <__private__/__pie_for_rounding.scad>;
 include <__private__/__half_trapezium.scad>;

src/rounded_extrude.scad

@@ -9,13 +9,13 @@
 **/
 
 include <__private__/__frags.scad>;
-include <__private__/__is_vector.scad>;
+include <__private__/__is_float.scad>;
 
 module rounded_extrude(size, round_r, angle = 90, twist = 0, convexity = 10) {
 
-    is_vt = __is_vector(size);
-    x = is_vt ? size[0] : size;
-    y = is_vt ? size[1] : size;
+    is_flt = __is_float(size);
+    x = is_flt ? size : size[0];
+    y = is_flt ? size : size[1];
     
     q_corner_frags = __frags(round_r) / 4;
     

src/rounded_square.scad

@@ -8,16 +8,16 @@
 *
 **/
 
-include <__private__/__is_vector.scad>;
+include <__private__/__is_float.scad>;
 include <__private__/__frags.scad>;
 include <__private__/__pie_for_rounding.scad>;
 include <__private__/__half_trapezium.scad>;
 include <__private__/__trapezium.scad>;
 
 module rounded_square(size, corner_r, center = false) {
-    is_vt = __is_vector(size);
-    x = is_vt ? size[0] : size;
-    y = is_vt ? size[1] : size;       
+    is_flt = __is_float(size);
+    x = is_flt ? size : size[0];
+    y = is_flt ? size : size[1];       
     
     position = center ? [0, 0] : [x / 2, y / 2];
     points = __trapezium(

src/shape_arc.scad

@@ -9,9 +9,10 @@
 **/ 
 
 include <__private__/__frags.scad>;
-include <__private__/__is_vector.scad>;
+include <__private__/__is_float.scad>;
 include <__private__/__ra_to_xy.scad>;
 include <__private__/__shape_arc.scad>;
+include <__private__/__edge_r.scad>
 
 function shape_arc(radius, angle, width, width_mode = "LINE_CROSS") =
     __shape_arc(radius, angle, width, width_mode);

src/shape_pie.scad

@@ -9,7 +9,7 @@
 **/
 
 include <__private__/__frags.scad>;
-include <__private__/__is_vector.scad>;
+include <__private__/__is_float.scad>;
 include <__private__/__ra_to_xy.scad>;
 include <__private__/__shape_pie.scad>;
 

src/shape_square.scad

@@ -8,17 +8,17 @@
 *
 **/
 
-include <__private__/__is_vector.scad>;
+include <__private__/__is_float.scad>;
 include <__private__/__frags.scad>;
 include <__private__/__pie_for_rounding.scad>;
 include <__private__/__half_trapezium.scad>;
 include <__private__/__trapezium.scad>;
-
+ 
 function shape_square(size, corner_r = 0) = 
     let(
-        is_vt = __is_vector(size),
-        x = is_vt ? size[0] : size,
-        y = is_vt ? size[1] : size        
+        is_flt = __is_float(size),
+        x = is_flt ? size : size[0],
+        y = is_flt ? size : size[1]        
     )
     __trapezium(
         length = x, 

src/shape_trapezium.scad

@@ -8,7 +8,7 @@
 *
 **/
 
-include <__private__/__is_vector.scad>;
+include <__private__/__is_float.scad>;
 include <__private__/__frags.scad>;
 include <__private__/__pie_for_rounding.scad>;
 include <__private__/__half_trapezium.scad>;

src/shear.scad

@@ -8,7 +8,6 @@
 *
 **/
 
-include <__private__/__m_multiply.scad>;
 include <__private__/__m_shearing.scad>;
 
 module shear(sx = [0, 0], sy = [0, 0], sz = [0, 0]) {

src/starburst.scad

@@ -0,0 +1,42 @@
+/**
+* starburst.scad
+*
+* @copyright Justin Lin, 2019
+* @license https://opensource.org/licenses/lgpl-3.0.html
+*
+* @see https://openhome.cc/eGossip/OpenSCAD/lib-starburst.html
+*
+**/
+
+module starburst(r1, r2, n, height) {
+    a = 180 / n;
+
+    p0 = [0, 0, 0];
+    p1 = [r2 * cos(a), r2 * sin(a), 0];
+    p2 = [r1, 0, 0];
+    p3 = [0, 0, height];
+
+    module half_burst() {
+        polyhedron(points = [p0, p1, p2, p3], 
+            faces = [
+                [0, 2, 1],
+                [0, 1, 3],
+                [0, 3, 2], 
+                [2, 1, 3]
+            ]
+        );
+    }
+
+    module burst() {
+        hull() {
+            half_burst();
+            mirror([0, 1,0]) half_burst();
+        }
+    }
+
+    union() {
+        for(i = [0 : n - 1]) {
+            rotate(2 * a * i) burst();
+        }
+    }
+}

src/torus_knot.scad

@@ -0,0 +1,21 @@
+/**
+* torus_knot.scad
+*
+* @copyright Justin Lin, 2019
+* @license https://opensource.org/licenses/lgpl-3.0.html
+*
+* @see https://openhome.cc/eGossip/OpenSCAD/lib-torus_knot.html
+*
+**/
+
+function torus_knot(p, q, phi_step) = [
+    for(phi = [0:phi_step:6.28318])
+    let(
+        degree = phi * 180 / 3.14159,
+        r = cos(q * degree) + 2,
+        x = r * cos(p * degree),
+        y = r * sin(p * degree),
+        z = -sin(q * degree)
+    )
+    [x, y, z]
+];

test/test_hull_polyline2d.scad

@@ -14,8 +14,6 @@ module test_hull_polyline2d() {
         assertEqualPoint(points[index], point2);
         assertEqual(line_width, radius * 2);    
     } 
-    
-    $fn = 4;
 
     hull_polyline2d(
         points = [[1, 2], [-5, -4], [-5, 3], [5, 5]],