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Concept
Line–line intersection
Summary
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.
Skew lines#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 .
First we consider the intersection of two lines L1 and L2 in two-dimensional space, with line L1 being defined by two distinct points (x1, y1) and (x2, y2), and line L2 being defined by two distinct points (x3, y3) and (x4, y4).
The intersection P of line L1 and L2 can be defined using determinants.
The determinants can be written out as:
When the two lines are parallel or coincident, the denominator is zero.
Intersection_(geometry)#Two_line_segments
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.
Official source
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