Line-line intersection

In Euclidean geometry, the intersection of a line and a line can be the empty set, a point, or a line. Distinguishing these cases and finding the intersection point have use, 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 are called skew lines and have no point of intersection. If they are in the same plane 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.

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 Skew lines#Testing for skewness.

First we consider the intersection of two lines ${\displaystyle L_{1}\,}$ and ${\displaystyle L_{2}\,}$ in 2-dimensional space, with line ${\displaystyle L_{1}\,}$ being defined by two distinct points ${\displaystyle (x_{1},y_{1})\,}$ and ${\displaystyle (x_{2},y_{2})\,}$, and line ${\displaystyle L_{2}\,}$ being defined by two distinct points ${\displaystyle (x_{3},y_{3})\,}$ and ${\displaystyle (x_{4},y_{4})\,}$.

...
Wikipedia