Drawing/Rendering 3D objects with epicycles and fourier transformations [Animation]

This answer is in response to: "Do you think [three.js] can replicate what i have in 2D but in 3D? with the rotating circles and stuff?"

Am not sure whether you're looking to learn 3D modeling from scratch (ie, creating your own library of vector routines, homogeneous coordinate transformations, rendering perspective, etc) or whether you're simply looking to produce a final product. In the case of the latter, three.js is a powerful graphics library built on webGL that in my estimation is simple enough for a beginner to dabble with, but has a lot of depth to produce very sophisticated 3D effects. (Peruse the examples at https://threejs.org/examples/ and you'll see for yourself.)

I happen to be working a three.js project of my own, and whipped up a quick example of epicyclic circles as a warm up exercise. This involved pulling pieces and parts from the following references...

  • https://threejs.org/docs/index.html#manual/en/introduction/Creating-a-scene

  • https://threejs.org/examples/#misc_controls_orbit

  • https://threejs.org/examples/#webgl_geometry_shapes (This three.js example is a great resource showing a variety of ways that a shape can be rendered.)

The result is a simple scene with one circle running around the other, permitting mouse controls to orbit around the scene, viewing it from different angles and distances.

    <title>Epicyclic Circles</title>
      body { margin: 0; }
      canvas { width: 100%; height: 100% }

    <script src="https://rawgit.com/mrdoob/three.js/dev/build/three.js"></script>
    <script src="https://rawgit.com/mrdoob/three.js/dev/examples/js/controls/OrbitControls.js"></script>


      // Set up the basic scene, camera, and lights.
      var scene = new THREE.Scene();
      scene.background = new THREE.Color( 0xf0f0f0 );
      var camera = new THREE.PerspectiveCamera( 75, window.innerWidth/window.innerHeight, 0.1, 1000 );
      var light = new THREE.PointLight( 0xffffff, 0.8 );
      camera.add( light );
      camera.position.z = 50;
      var renderer = new THREE.WebGLRenderer();
      renderer.setSize( window.innerWidth, window.innerHeight );
      document.body.appendChild( renderer.domElement );
      // Add the orbit controls to permit viewing the scene from different angles via the mouse.
      controls = new THREE.OrbitControls( camera, renderer.domElement );
      controls.enableDamping = true; // an animation loop is required when either damping or auto-rotation are enabled
      controls.dampingFactor = 0.25;
      controls.screenSpacePanning = false;
      controls.minDistance = 0;
      controls.maxDistance = 500;
      // Create center and epicyclic circles, extruding them to give them some depth.
      var extrudeSettings = { depth: 2, bevelEnabled: true, bevelSegments: 2, steps: 2, bevelSize: .25, bevelThickness: .25 };

      var arcShape1 = new THREE.Shape();
      arcShape1.moveTo( 0, 0 );
      arcShape1.absarc( 0, 0, 15, 0, Math.PI * 2, false );
      var holePath1 = new THREE.Path();
      holePath1.moveTo( 0, 10 );
      holePath1.absarc( 0, 10, 2, 0, Math.PI * 2, true );
      arcShape1.holes.push( holePath1 );
      var geometry1 = new THREE.ExtrudeBufferGeometry( arcShape1, extrudeSettings );
      var mesh1 = new THREE.Mesh( geometry1, new THREE.MeshPhongMaterial( { color: 0x804000 } ) );
      scene.add( mesh1 );
      var arcShape2 = new THREE.Shape();
      arcShape2.moveTo( 0, 0 );
      arcShape2.absarc( 0, 0, 15, 0, Math.PI * 2, false );
      var holePath2 = new THREE.Path();
      holePath2.moveTo( 0, 10 );
      holePath2.absarc( 0, 10, 2, 0, Math.PI * 2, true );
      arcShape2.holes.push( holePath2 );
      var geometry2 = new THREE.ExtrudeGeometry( arcShape2, extrudeSettings );
      var mesh2 = new THREE.Mesh( geometry2, new THREE.MeshPhongMaterial( { color: 0x00ff00 } ) );
      scene.add( mesh2 );

      // Define variables to hold the current epicyclic radius and current angle.
      var mesh2AxisRadius = 30;
      var mesh2AxisAngle = 0;

      var animate = function () {
        requestAnimationFrame( animate );

        // During each animation frame, let's rotate the objects on their center axis,  
        // and also set the position of the epicyclic circle.
        mesh1.rotation.z -= 0.02;

        mesh2.rotation.z += 0.02;
        mesh2AxisAngle += 0.01;
        mesh2.position.set ( mesh2AxisRadius * Math.cos(mesh2AxisAngle), mesh2AxisRadius * Math.sin(mesh2AxisAngle), 0 );

        renderer.render( scene, camera );


Note that I've used basic trigonometry within the animate function to position the epicyclic circle around the center circle, and fudged the rate of rotation for the circles (rather than doing the precise math), but there's probably a better "three.js"-way of doing this via matrices or built in functions. Given that you obviously have a strong math background, I don't think you'll have any issues with translating your 2D model of multi-epicyclic circles using basic trigonometry when porting to 3D.

Hope this helps in your decision making process on how to proceed with a 3D version of your program.

The method that I would suggest is as follows. Start with a parametrized path v(t) = (v_x(t), v_y(t), v_z(t)). Consider the following projection onto the X-Y plane: v1(t) = (v_x(t)/2, v_y(t), 0). And the corresponding projection onto the X-Z plane: v2(t) = (v_x(t)/2, 0, v_z(t)).

When we add these projections together we get the original curve. But each projection is now a closed 2-D curve, and you have solutions for arbitrary closed 2-D curves. So solve each problem. And then interleave them to get a projection where your first circle goes in the X-Y plane, your second one in the X-Z plane, your third one in the X-Y plane, your fourth one in the X-Z plane ... and they sum up to your answer!