Radius of projected sphere in screen space
The illustrated accepted answer above is excellent, but I needed a solution without knowing the field of view, just a matrix to transform between world and screen space, so I had to adapt the solution.
Reusing some variable names from the other answer, calculate the start point of the spherical cap (the point where line
h
meets lined
):capOffset = cos(asin(l / d)) * r capCenter = sphereCenter + ( sphereNormal * capOffset )
where
capCenter
andsphereCenter
are points in world space, andsphereNormal
is a normalized vector pointing alongd
, from the sphere center towards the camera.Transform the point to screen space:
capCenter2 = matrix.transform(capCenter)
Add
1
(or any amount) to thex
pixel coordinate:capCenter2.x += 1
Transform it back to world space:
capCenter2 = matrix.inverse().transform(capCenter2)
Measure the distance between the original and new points in world space, and divide into the amount you added to get a scale factor:
scaleFactor = 1 / capCenter.distance(capCenter2)
Multiply that scale factor by the cap radius
h
to get the visible screen radius in pixels:screenRadius = h * scaleFactor
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 spherer
: radius of the spherel
: 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 eyephi
: 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>
Let the sphere have radius r
and be seen at a distance d
from the observer. The projection plane is at distance f
from the observer.
The sphere is seen under the half angle asin(r/d)
, so the apparent radius is f.tan(asin(r/d))
, which can be written as f . r / sqrt(d^2 - r^2)
. [The wrong formula being f . r / d
.]