Line–sphere intersection

In analytic geometry, a line and a sphere can intersect in three ways: 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.

[1] 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: