Line-plane intersection

In analytic geometry, the intersection of a line and a plane in three-dimensional space can be the empty set, a point, or a line. It is the entire line if that line is embedded in the plane, and is the empty set if the line is parallel to the plane but outside it. Otherwise, the line cuts through the plane at a single point.

Distinguishing these cases, and determining equations for the point and line in the latter cases, have use in computer graphics, motion planning, and collision detection.

In vector notation, a plane can be expressed as the set of points ${\displaystyle \mathbf {p} }$ for which

where ${\displaystyle \mathbf {n} }$ is a normal vector to the plane and ${\displaystyle \mathbf {p_{0}} }$ is a point on the plane. (The notation ${\displaystyle \mathbf {a} \cdot \mathbf {b} }$ denotes the dot product of the vectors ${\displaystyle \mathbf {a} }$ and ${\displaystyle \mathbf {b} }$.)

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