From a0c56ceb45626371e81f2f717e4e92604a655d7a Mon Sep 17 00:00:00 2001
From: dakkar <dakkar@thenautilus.net>
Date: Sun, 2 Feb 2020 19:25:56 +0000
Subject: comments

---
 ambiguous-cylinders.scad | 75 ++++++++++++++++++++++++++++++++++++++++++++++--
 1 file changed, 72 insertions(+), 3 deletions(-)

diff --git a/ambiguous-cylinders.scad b/ambiguous-cylinders.scad
index 9a59522..965942d 100644
--- a/ambiguous-cylinders.scad
+++ b/ambiguous-cylinders.scad
@@ -1,3 +1,25 @@
+/*
+  Recreating Sugihara's impossible objects
+
+  Original work: http://www.isc.meiji.ac.jp/~kokichis/Welcomee.html
+
+  PDF Paper with the maths: https://www.mdpi.com/2073-8994/8/4/21
+
+  Based on Sugihara's work, I've approximated his mathematical model
+  with OpenSCAD.
+ */
+
+/* let's build it in steps:
+
+   given a 2D object, extrude a `thin` "cylinder" with its shape as
+   section; this approximates the set of points that would appear
+   indistinguishable from the given shape to an observer infinitely
+   distant on the Z axis
+
+   (`height` determines how tall the cylinder is, `shift` determines
+   how low it is on Z axis)
+*/
+
 module view_cylinder(shift=100/3,height=200,thin=0.1) {
      translate([0,0,-shift]) {
           difference() {
@@ -11,6 +33,24 @@ module view_cylinder(shift=100/3,height=200,thin=0.1) {
      }
 }
 
+/*
+  given two objects:
+
+  * build they view cylinders
+
+  * rotate them so the two "observers" are opposite each other on the
+    Y axis, lookin toward the origin with a downward `angle`
+
+  * intersect them
+
+  the result approximates the set of points that would look like the
+  first object to the first observers, and like the second object to
+  the second observer
+
+  (`size` should be higher than the diameter of the largest child,
+  otherwise the shapes may get cut)
+*/
+
 module intersect_view_cylinders(angle=45,size=100,thin=0.1) {
      shift=size/3;
      height=size*2;
@@ -21,6 +61,21 @@ module intersect_view_cylinders(angle=45,size=100,thin=0.1) {
      }
 };
 
+/*
+  `intersect_view_cylinders` produces a slightly ugly shape, with two
+  half-curves for each half-object; to see the problem, render this:
+
+intersect_view_cylinders() {
+     circle(d=14);
+     rotate([0,0,45]) square(size=10,center=true);
+}
+
+  therefore, we render it twice, cutting the "wrong half" out each
+  time. This gives us the curve we want at the top of our ambiguous
+  cylinder
+*/
+
+
 module clip_intersect_view(angle=45,size=100,thin=0.1) {
      bound=size*2;
      skip=0;
@@ -41,17 +96,30 @@ module clip_intersect_view(angle=45,size=100,thin=0.1) {
      }
 };
 
+/*
+  finally, we "extrude" the curve; OpenSCAD can't extrude a 3d shape,
+  but we can "paint" it with a vertical cylinder, and that's close
+  enough
+
+  (`thick` determines the thickness of the wall, `height` its height)
+ */
+
 module ambiguous_cylinder(angle=45,size=100,thin=0.1,thick=1,height=10) {
      minkowski() {
           clip_intersect_view(angle,size,thin) {
                children(0);
                children(1);
           }
-          cylinder(r=thick,h=height*2,center=true);
+          cylinder(r=thick/2,h=height*2,center=true);
      }
 }
 
-/* Arrow that always points right
+/*
+  Some examples:
+ */
+
+/*
+  Arrow that always points right
   */
 module arrow(length=10) {
      scale([length/10,length/10,length/10])
@@ -64,7 +132,8 @@ ambiguous_cylinder() {
      rotate([0,0,-90]) arrow(30);
 }
 
-/* circle / diamond
+/*
+  circle / diamond
  */
 
 translate([0,50,0]) ambiguous_cylinder() {
-- 
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