In geometry, the intersection of a line and a plane in three-dimensional space can be the empty set, a point, or the line itself. 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.

Algebraic form

In vector notation for which

is a normal vector denotes the dot product.)

The vector equation for a line is

is a unit vector is a scalar in the real number domain. Substituting the equation for the line into the equation for the plane gives

Expanding gives

gives

then the line is contained in the plane, that is, the line intersects the plane at each point of the line. Otherwise, the line and plane have no intersection.

, is given by

1. .

Parametric form

can be represented as

.

can be represented as

.

The point at which the line intersects the plane is therefore described by setting the point on the line equal to the point on the plane, giving the parametric equation:

This can be rewritten as

which can be expressed in matrix form as

where the vectors are written as column vectors.

This produces a system of linear equations, then the intersection point is in the parallelogram.

The determinant of the matrix can be calculated as

If the determinant is zero, then there is no unique solution; the line is either in the plane or parallel to it.

If a unique solution exists (determinant is not 0), then it can be found by inverting the matrix and rearranging:

which expands to

and then to

thus giving the solutions:

The point of intersection is then equal to

Uses

In the ray tracing method of computer graphics a surface can be represented as a set of pieces of planes. The intersection of a ray of light with each plane is used to produce an image of the surface. In vision-based 3D reconstruction, a subfield of computer vision, depth values are commonly measured by so-called triangulation method, which finds the intersection between light plane and ray reflected toward camera.

The algorithm can be generalised to cover intersection with other planar figures, in particular, the intersection of a polyhedron with a line.

See also

• calculating the intersection when the line is expressed by Plücker coordinates.

• Plane–plane intersection

References

• Intersection of a Line and a Plane

cs:Analytická geometrie#Vzájemná poloha dvou rovin v třírozměrném prostoru