Line–sphere intersection


In analytic geometry, a line and a sphere can intersect in three ways:
  1. no intersection at all,
  2. intersection at exactly one point, and
  3. intersection at two points.
Methods for distinguishing these cases, and determining the coordinates for the points in the latter cases, are useful in a number of circumstances. For example, it is a common calculation to perform during ray tracing.

Calculation using vectors in 3D

In vector notation, the equations are as follows:
Equation for a sphere
Equation for a line starting at
Searching for points that are on the line and on the sphere means combining the equations and solving for, involving the dot product of vectors: