Radius of projected sphere in screen space

I'm trying to find the visible size of a sphere in pixels, after projection to screen space. The sphere is centered at the origin with the camera looking right at it. Thus the projected sphere should be a perfect circle in two dimensions. I am aware of this 1 existing question. However, the formula given there doesn't seem to produce the result I want. It is too small by a few percent. I assume this is because it is not correctly taking perspective into account. After projecting to screen space you do not see half the sphere but significantly less, due to perspective foreshortening (you see just a cap of the sphere instead of the full hemisphere 2).

How can I derive an exact 2D bounding circle?

Indeed, with a perspective projection you need to compute the height of the sphere "horizon" from the eye / center of the camera (this "horizon" is determined by rays from the eye tangent to the sphere).

Notations:

d: distance between the eye and the center of the sphere
r: radius of the sphere
l: distance between the eye and a point on the sphere "horizon", l = sqrt(d^2 - r^2)
h: height / radius of the sphere "horizon"
theta: (half-)angle of the "horizon" cone from the eye
phi: complementary angle of theta

h / l = cos(phi)

but:

r / d = cos(phi)

so, in the end:

h = l * r / d = sqrt(d^2 - r^2) * r / d

Then once you have h, simply apply the standard formula (the one from the question you linked) to get the projected radius pr in the normalized viewport:

pr = cot(fovy / 2) * h / z

with z the distance from the eye to the plane of the sphere "horizon":

z = l * cos(theta) = sqrt(d^2 - r^2) * h / r

so:

pr = cot(fovy / 2) * r / sqrt(d^2 - r^2)

And finally, multiply pr by height / 2 to get the actual screen radius in pixels.

What follows is a small demo done with three.js. The sphere distance, radius and the vertical field of view of the camera can be changed by using respectively the n / f, m / p and s / w pairs of keys. A yellow line segment rendered in screen-space shows the result of the computation of the radius of the sphere in screen-space. This computation is done in the function computeProjectedRadius().

projected-sphere.js:

"use strict";

function computeProjectedRadius(fovy, d, r) {
  var fov;

  fov = fovy / 2 * Math.PI / 180.0;

//return 1.0 / Math.tan(fov) * r / d; // Wrong
  return 1.0 / Math.tan(fov) * r / Math.sqrt(d * d - r * r); // Right
}

function Demo() {
  this.width = 0;
  this.height = 0;

  this.scene = null;
  this.mesh = null;
  this.camera = null;

  this.screenLine = null;
  this.screenScene = null;
  this.screenCamera = null;

  this.renderer = null;

  this.fovy = 60.0;
  this.d = 10.0;
  this.r = 1.0;
  this.pr = computeProjectedRadius(this.fovy, this.d, this.r);
}

Demo.prototype.init = function() {
  var aspect;
  var light;
  var container;

  this.width = window.innerWidth;
  this.height = window.innerHeight;

  // World scene
  aspect = this.width / this.height;
  this.camera = new THREE.PerspectiveCamera(this.fovy, aspect, 0.1, 100.0);

  this.scene = new THREE.Scene();
  this.scene.add(THREE.AmbientLight(0x1F1F1F));

  light = new THREE.DirectionalLight(0xFFFFFF);
  light.position.set(1.0, 1.0, 1.0).normalize();
  this.scene.add(light);

  // Screen scene
  this.screenCamera = new THREE.OrthographicCamera(-aspect, aspect,
                                                   -1.0, 1.0,
                                                   0.1, 100.0);
  this.screenScene = new THREE.Scene();

  this.updateScenes();

  this.renderer = new THREE.WebGLRenderer({
    antialias: true
  });
  this.renderer.setSize(this.width, this.height);
  this.renderer.domElement.style.position = "relative";
  this.renderer.autoClear = false;

  container = document.createElement('div');
  container.appendChild(this.renderer.domElement);
  document.body.appendChild(container);
}

Demo.prototype.render = function() {
  this.renderer.clear();
  this.renderer.setViewport(0, 0, this.width, this.height);
  this.renderer.render(this.scene, this.camera);
  this.renderer.render(this.screenScene, this.screenCamera);
}

Demo.prototype.updateScenes = function() {
  var geometry;

  this.camera.fov = this.fovy;
  this.camera.updateProjectionMatrix();

  if (this.mesh) {
    this.scene.remove(this.mesh);
  }

  this.mesh = new THREE.Mesh(
    new THREE.SphereGeometry(this.r, 16, 16),
    new THREE.MeshLambertMaterial({
      color: 0xFF0000
    })
  );
  this.mesh.position.z = -this.d;
  this.scene.add(this.mesh);

  this.pr = computeProjectedRadius(this.fovy, this.d, this.r);

  if (this.screenLine) {
    this.screenScene.remove(this.screenLine);
  }

  geometry = new THREE.Geometry();
  geometry.vertices.push(new THREE.Vector3(0.0, 0.0, -1.0));
  geometry.vertices.push(new THREE.Vector3(0.0, -this.pr, -1.0));

  this.screenLine = new THREE.Line(
    geometry,
    new THREE.LineBasicMaterial({
      color: 0xFFFF00
    })
  );

  this.screenScene = new THREE.Scene();
  this.screenScene.add(this.screenLine);
}

Demo.prototype.onKeyDown = function(event) {
  console.log(event.keyCode)
  switch (event.keyCode) {
    case 78: // 'n'
      this.d /= 1.1;
      this.updateScenes();
      break;
    case 70: // 'f'
      this.d *= 1.1;
      this.updateScenes();
      break;
    case 77: // 'm'
      this.r /= 1.1;
      this.updateScenes();
      break;
    case 80: // 'p'
      this.r *= 1.1;
      this.updateScenes();
      break;
    case 83: // 's'
      this.fovy /= 1.1;
      this.updateScenes();
      break;
    case 87: // 'w'
      this.fovy *= 1.1;
      this.updateScenes();
      break;
  }
}

Demo.prototype.onResize = function(event) {
  var aspect;

  this.width = window.innerWidth;
  this.height = window.innerHeight;

  this.renderer.setSize(this.width, this.height);

  aspect = this.width / this.height;
  this.camera.aspect = aspect;
  this.camera.updateProjectionMatrix();

  this.screenCamera.left = -aspect;
  this.screenCamera.right = aspect;
  this.screenCamera.updateProjectionMatrix();
}

function onLoad() {
  var demo;

  demo = new Demo();
  demo.init();

  function animationLoop() {
    demo.render();
    window.requestAnimationFrame(animationLoop);
  }

  function onResizeHandler(event) {
    demo.onResize(event);
  }

  function onKeyDownHandler(event) {
    demo.onKeyDown(event);
  }

  window.addEventListener('resize', onResizeHandler, false);
  window.addEventListener('keydown', onKeyDownHandler, false);
  window.requestAnimationFrame(animationLoop);
}

index.html:

<!DOCTYPE html>
<html>
  <head>
    <title>Projected sphere</title>
      <style>
        body {
            background-color: #000000;
        }
      </style>
      <script src="http://cdnjs.cloudflare.com/ajax/libs/three.js/r61/three.min.js"></script>
      <script src="projected-sphere.js"></script>
    </head>
    <body onLoad="onLoad()">
      <div id="container"></div>
    </body>
</html>