What is the method for determining the angle between two planes?

My question is about calculating the angle between two planes. Is it also possible to calculate the angle between two Object3D points like we do with planes?

If you need a visual example, check out this fiddle: https://jsfiddle.net/rsu842v8/1/

  const scene = new THREE.Scene();
  const camera = new THREE.PerspectiveCamera(45, window.innerWidth / window.innerHeight, 1, 1000);
  camera.position.set(25, 25, 12);

  var material = new THREE.MeshBasicMaterial({
    color: 0x00fff0,
    side: THREE.DoubleSide
  });
  window.plane1 = new THREE.Mesh(new THREE.PlaneGeometry(10, 10), material);
  scene.add(plane1);

  plane1.position.set(0.3, 1, -2);
  plane1.rotation.set(Math.PI / 3, Math.PI / 2, 1);

  window.plane2 = new THREE.Mesh(new THREE.PlaneGeometry(10, 10), new THREE.MeshBasicMaterial({
    color: 0x0fff00,
    side: THREE.DoubleSide
  }));
  scene.add(plane2);

  // setup lighting
  var pointLight = new THREE.PointLight(0xFFFFFF);
  pointLight.position.x = 10;
  pointLight.position.y = 50;
  pointLight.position.z = 130;
  scene.add(pointLight)

  const renderer = new THREE.WebGLRenderer({
    antialias: true
  });
  renderer.setSize(window.innerWidth, window.innerHeight);
  renderer.setClearColor(0x20252f);
  renderer.setPixelRatio(window.devicePixelRatio);
  document.body.appendChild(renderer.domElement);

  const controls = new THREE.OrbitControls(camera, renderer.domElement);

  animate();

  // TODO: What is the angle between plane1 and plane2?

  function animate() {
    requestAnimationFrame(animate);
    render();
  }

  function render() {
    renderer.render(scene, camera);
  }
<script src="https://cdnjs.cloudflare.com/ajax/libs/three.js/r82/three.js"></script>
<script src="https://yume.human-interactive.org/examples/buffer-geometry/OrbitControls.js"></script>

Answer №1

When working with three.js plane meshes, one common task is finding the angle between them.

By default, a THREE.PlaneGeometry is aligned to face the positive z-axis without any rotation. This means that the normal of the plane points towards the positive z-axis.

To determine the angle between two planes, you can create a ( 0, 0, 1 ) vector and apply the same rotation transformation as done to the plane meshes.

Keep in mind that when adjusting the rotation of a plane using plane.rotation, the quaternion value at plane.quaternion gets automatically updated. You can utilize this quaternion for the angle calculation like this:

var vec1 = new THREE.Vector3( 0, 0, 1 ); // initialize once and reuse
var vec2 = new THREE.Vector3( 0, 0, 1 );

vec1.applyQuaternion( plane1.quaternion );
vec2.applyQuaternion( plane2.quaternion );

var angle = vec1.angleTo( vec2 ); // in radians

If the planes are nested within other rotated objects, the problem becomes more complex.

Additionally, the angleTo() function can be used to find the angle between any two vectors.

This explanation pertains to three.js version r.86.

Answer №2

To find the angle between two planes you are rendering, it's recommended to calculate the normal vectors for each plane. Once you have these vectors - let's call them n1 and n2 - you can easily determine the angle between the planes using the dot product.

If you're not familiar with the dot product, dot(n1,n2) where n1 = (x1,y1,z1) and n2 = (x2,y2,z2) would be equal to x1*x2 + y1*y2 + z1*z2. Another identity states that dot(n1,n2) = |v1||v2|cos(a) where || represents the magnitude of a vector - for example, |v| = sqrt(x*x + y*y + z*z) if v = (x,y,z) - and 'a' is the angle between the normals which is essentially the angle between the planes. You may refer to this Mathematics Stack Exchange answer for more information.

In summary,

a = arccos(dot(n1,n2) / |n1||n2|)
.

If you already know that n1 and n2 are unit vectors, then the equation simplifies to a = arccos(dot(n1,n2)).

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