K_{3,3} embedded on a Möbius strip (with Bezier curve edges)

utility.jscad

"use strict";
const jscad = require("@jscad/modeling");
const { vec2, vec3} = jscad.maths;
const { bezier} = jscad.curves;
const { rotate, scale, translate } = jscad.transforms;
const { cuboid, sphere, cylinder } = jscad.primitives;
const { colorize } = jscad.colors;

const r=40;
const w=20;
const t=0.001;

let edgeCylinder = cylinder({height: 1, radius: 0.3});  // cached

function edge(v, w) {
    let d = vec3.subtract(vec3.create(), w, v);
    let m = vec3.lerp(vec3.create(), v, w, 0.5);
    let R = [0, Math.acos(d[2] / vec3.length(d)), Math.atan2(d[1], d[0])];
    let S = [1, 1, vec3.length(d)];
    return translate(m, rotate(R, scale(S, edgeCylinder)));
}
function vertex(c) {
    return translate(c, sphere({radius: 1.6}));
}

function xyz(a) {
  return vec3.rotateZ(vec3.create(), 
                      vec3.add(vec3.create(),
                               vec3.rotateY(vec3.create(), 
                                            [0,0,a[1]],
                                            [0, 0, 0],
                                            a[0]/2),
                               [r, 0, 0]),
                      [0, 0, 0],
                      a[0]);
}

function moebius_edge(a) {
    const n = 100;
    let ret = [];
    let from = a[0];
    let i = 1;
    let b = bezier.create(a);
    while (i <= n) {
        let to = bezier.valueAt(i / n, b);
        ret.push(edge(xyz(from), xyz(to)));
        from = to;
        i += 1;
    }
    return ret;
}

function moebius_strip(r, w, t) {
  var ret = [];
  var step = 0.5/180.0*Math.PI;
  var azimuth;
  var x;
  var y;
  for(azimuth = 0; azimuth <= 2*Math.PI; azimuth += step) {
    x = r * Math.cos(azimuth);
    y = r * Math.sin(azimuth);
    ret.push(translate([x, y, 0],rotate([0, azimuth/2, azimuth],cuboid({size: [t, 128*step, w]}))));
  }
  return ret;
}

function main() {
  let Vs = [
    [13*Math.PI/8,-3],
    [14*Math.PI/8,-3],
    [15*Math.PI/8,-3],
    [13*Math.PI/8,3],
    [14*Math.PI/8,3],
    [15*Math.PI/8,3],
    [12*Math.PI/8,13],
    [15.9*Math.PI/8,13],
    [15.9*Math.PI/8,-16],
    [14*Math.PI/8,-11],
    [0*Math.PI/8,-4],
    [0*Math.PI/8,7],
    [2*Math.PI-0.0001,4],
    [2*Math.PI-0.0001,-7],
  ];
  let Es = [[1, 3], [1, 4], [1, 5], [0, 3], [2, 5],
            [0, 6, 4],
            [2, 7, 4],
            [3, 8, 10],
            [5, 9, 13],
            [0, 11], [2, 12]
           ];

  var out = [];
  Vs.forEach(function(v,i){
    if(i<6)
      out.push(colorize([1,0.5,0],vertex(xyz(v))))
  })
  Es.forEach(function (e) {
   out.push(colorize([1, 1, 0], moebius_edge(e.map(function (v){return Vs[v]}))));
  });
  return [
    colorize([0.66, 0.0, 0.66],moebius_strip(r,w,t)),
    out,
  ];
}

module.exports = { main };

JSCAD app link to play with. Below view when clicking on corner TOP/FRONT/RIGHT.

JSCAD app link to play with for new revision with Bezier cuerve edges.
Below view when clicking on corner TOP/FRONT/RIGHT.