In Euclidean geometry, the intersection of a line and a line can be the empty set, a point, or another line. Distinguishing these cases and finding the intersection have uses, for example, in computer graphics, motion planning, and collision detection.
In three-dimensional Euclidean geometry, if two lines are not in the same plane, they have no point of intersection and are called skew lines. If they are in the same plane, however, there are three possibilities: if they coincide (are not distinct lines), they have an infinitude of points in common (namely all of the points on either of them); if they are distinct but have the same slope, they are said to be parallel and have no points in common; otherwise, they have a single point of intersection.
The distinguishing features of non-Euclidean geometry are the number and locations of possible intersections between two lines and the number of possible lines with no intersections (parallel lines) with a given line.
Formulas
A necessary condition for two lines to intersect is that they are in the same plane—that is, are not skew lines. Satisfaction of this condition is equivalent to the tetrahedron with vertices at two of the points on one line and two of the points on the other line being degenerate in the sense of having zero volume. For the algebraic form of this condition, see .
Given two points on each line
First we consider the intersection of two lines and in two-dimensional space, with line being defined by two distinct points and , and line being defined by two distinct points and .
The intersection of line and can be defined using determinants.
The determinants can be written out as:
When the two lines are parallel or coincident, the denominator is zero.
Given two points on each line segment
The intersection point above is for the infinitely long lines defined by the points, rather than the line segments between the points, and can produce an intersection point not contained in either of the two line segments. In order to find the position of the intersection in respect to the line segments, we can define lines and in terms of first degree Bézier parameters:
(where and are real numbers). The intersection point of the lines is found with one of the following values of or , where
and
with
There will be an intersection if and . The intersection point falls within the first line segment if , and it falls within the second line segment if . These inequalities can be tested without the need for division, allowing rapid determination of the existence of any line segment intersection before calculating its exact point.
Given two line equations
The and coordinates of the point of intersection of two non-vertical lines can easily be found using the following substitutions and rearrangements.
Suppose that two lines have the equations and where and are the slopes (gradients) of the lines and where and are the -intercepts of the lines. At the point where the two lines intersect (if they do), both coordinates will be the same, hence the following equality:
We can rearrange this expression in order to extract the value of ,
and so,
To find the coordinate, all we need to do is substitute the value of into either one of the two line equations, for example, into the first:
Hence, the point of intersection is
Note if then the two lines are parallel. If as well, the lines are different and there is no intersection, otherwise the two lines are identical and intersect at every point.
Using homogeneous coordinates
By using homogeneous coordinates, the intersection point of two implicitly defined lines can be determined quite easily. In 2D, every point can be defined as a projection of a 3D point, given as the ordered triple . The mapping from 3D to 2D coordinates is . We can convert 2D points to homogeneous coordinates by defining them as .
Assume that we want to find intersection of two infinite lines in 2-dimensional space, defined as and . We can represent these two lines in line coordinates as and . The intersection of two lines is then simply given by
If , the lines do not intersect.
More than two lines
The intersection of two lines can be generalized to involve additional lines. The existence of and expression for the -line intersection problem are as follows.
In two dimensions
In two dimensions, more than two lines almost certainly do not intersect at a single point. To determine if they do and, if so, to find the intersection point, write the th equation () as