In analytic
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
, the intersection of a
line and a
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ...
in
three-dimensional space
Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informa ...
can be the
empty set, a
point
Point or points may refer to:
Places
* Point, Lewis, a peninsula in the Outer Hebrides, Scotland
* Point, Texas, a city in Rains County, Texas, United States
* Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland
* 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
Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great de ...
,
motion planning
Motion planning, also path planning (also known as the navigation problem or the piano mover's problem) is a computational problem to find a sequence of valid configurations that moves the object from the source to destination. The term is use ...
, and
collision detection
Collision detection is the computational problem of detecting the intersection of two or more objects. Collision detection is a classic issue of computational geometry and has applications in various computing fields, primarily in computer grap ...
.
Algebraic form
In
vector notation
In mathematics and physics, vector notation is a commonly used notation for representing vectors, which may be Euclidean vectors, or more generally, members of a vector space.
For representing a vector, the common typographic convention is l ...
, 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 notation
denotes the
dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alge ...
of the vectors
and
.)
The vector equation for a line is
:
where
is a vector in the direction of the line,
is a point on the line, and
is a scalar in the
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
domain. Substituting the equation for the line into the equation for the plane gives
:
Expanding gives
:
And solving for
gives
:
If
then the line and plane are parallel. There will be two cases: if
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.
If
there is a single point of intersection. The value of
can be calculated and the point of intersection,
, is given by
:
.
Parametric form
A line is described by all points that are a given direction from a point. A general point on a line passing through points
and
can be represented as
:
where
is the vector pointing from
to
.
Similarly a general point on a plane determined by the triangle defined by the points
,
and
can be represented as
:
where
is the vector pointing from
to
, and
is the vector pointing from
to
.
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 which can be solved for
,
and
. If the solution satisfies the condition
, then the intersection point is on the line segment between
and
, otherwise it is elsewhere on the line. Likewise, if the solution satisfies
, then the intersection point is in the
parallelogram formed by the point
and vectors
and
. If the solution additionally satisfies
, then the intersection point lies in the triangle formed by the three points
,
and
.
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
Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great de ...
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
*
Plücker coordinates#Plane-line meet calculating the intersection when the line is expressed by Plücker coordinates.
*
Plane–plane intersection
External links
Intersections of Lines, Segments and Planes (2D & 3D) from GeomAlgorithms.com
{{DEFAULTSORT:Line-Plane Intersection
Euclidean geometry
Computational physics
Geometric algorithms
Geometric intersection
cs:Analytická geometrie#Vzájemná poloha dvou rovin v třírozměrném prostoru