### Radius of a Projected Sphere

It's useful to calculate sphere projection in clip space. One usage is to determine LOD of a model. As you might have known, sphere projection results in ellipse instead of sphere. In this post, a way to approximate radius of projected sphere is given.

Assume the sphere is in the center of the screen and we know the \(FOV\) of the camera, sphere radius \(r\) and distance from camera \(z\). Looking at the diagram below:

The approximate projection of sphere radius in clip space is given by:

$$ approx\_projected\_radius = \frac{y}{y_{max}} = \frac{r}{z \times \tan (\frac{FOV}{2})} $$

Assume the sphere is in the center of the screen and we know the \(FOV\) of the camera, sphere radius \(r\) and distance from camera \(z\). Looking at the diagram below:

The approximate projection of sphere radius in clip space is given by:

$$ approx\_projected\_radius = \frac{y}{y_{max}} = \frac{r}{z \times \tan (\frac{FOV}{2})} $$