In analytic geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, the intersection of a line

Line (mathematics)

The notion of line or straight line was introduced by the ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects...

 and a plane

Plane (mathematics)

In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...

 can be the empty set

Empty set

In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...

,
a point

Point (geometry)

In geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...

, or
a line. Distinguishing these cases, and determining equations for the point and line in the latter cases have use, for example, in computer graphics

Computer graphics

Computer graphics are graphics created using computers and, more generally, the representation and manipulation of image data by a computer with help from specialized software and hardware....

, motion planning

Motion planning

Motion planning is a term used in robotics for the process of detailing a task into discrete motions....

, and collision detection

Collision detection

Collision detection typically refers to the computational problem of detecting the intersection of two or more objects. While the topic is most often associated with its use in video games and other physical simulations, it also has applications in robotics...

.

Parametric form

A line is described by all points that are a given direction from a point. Thus a general point on a line can be represented as


where and
are two distinct points along the line.

Similarly a general point on a plane can be represented as


where ,
are three points in the plane which are not co-linear.

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 simplified to
which can be expressed in matrix form as:

The point of intersection is then equal to

If the line is parallel to the plane then the vectors , , and will be linearly dependent and the matrix will be singular. This situation will also occur when the line lies in the plane.

If the solution satisfies the condition , then the intersection point is on the line between and .

If the solution satisfies
then the intersection point is in the plane inside the triangle spanned by the three points , and .

This problem is typically solved by expressing it in matrix form, and inverting it:

Algebraic form

In vector notation

Vector notation

This page is an overview of the common notations used when working with vectors, which may be spatial or more abstract members of vector spaces....

, a plane can be expressed as the set of points for which
where is a normal vector to the plane and is a point on the plane.
The vector equation for a line is
where is a vector in the direction of the line and is a point on the line. Substitute the line into the plane equation to get
Distribute to get
And solve for

If the line starts outside the plane and is parallel to the plane, there is no intersection. In this case, the above denominator will be zero and the numerator will be non-zero. If the line starts inside the plane and is parallel to the plane, the line intersects the plane everywhere. In this case, both the numerator and denominator above will be zero. In all other cases, the line intersects the plane once and represents the intersection as the distance along the line from .

Uses

In the ray tracing method of computer graphics

Computer graphics

Computer graphics are graphics created using computers and, more generally, the representation and manipulation of image data by a computer with help from specialized software and hardware....

 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

Intersection of a polyhedron with a line

In computational geometry, the intersection of a polyhedron with a line is the problem of computing the intersection of a convex polyhedron and a ray in Euclidean space...

.