An ellipsoid SDF is computed by scaling space to transform the ellipsoid into a unit sphere

The SDF for a perfect sphere is length(p) - r. An ellipsoid with three different radii (rx, ry, rz) can be evaluated by dividing each component of p by its corresponding radius before computing length, then dividing the result by the minimum radius to approximately recover a Euclidean distance. The exact ellipsoid SDF requires iterative solving, but the space-distortion approach is cheaper and accurate enough for most rendering purposes. The distortion causes the field to be slightly non-Euclidean for very elongated ellipsoids.

Examples

float sdEllipsoid(vec3 p, vec3 r){ vec3 q=p/r; return (length(q)-1.0)*min(min(r.x,r.y),r.z); }

Assessment

Why does the min(r.x,r.y,r.z) factor in the ellipsoid SDF approximation help, and when does it fail (what shape produces the worst error)?