Intersection (Euclidean geometry)

In geometry, an intersection is a point, line, or curve common in two or more objects (such as lines, curves, planes, and surfaces). The most simple case in Euclidean geometry is the intersection points of two distinct lines, that is either one point or does not exist if lines are parallel.

Determination of the intersection of flats is a simple task of linear algebra, namely a system of linear equations. In general the determination of an intersection leads to non-linear equations, which can be solved numerically, for example using a Newton iteration. Intersection problems between a line and a conic section (circle, ellipse, parabola, ...) or a quadric (sphere, cylinder, hyperboloid, ...) lead to quadratic equations that can be easily solved. Intersections between quadrics lead to quartic equations that can be solved algebraically.

On a plane

Two lines

For the determination of the intersection point of two non-parallel lines

one gets from Cramer's rule for the coordinates of the intersection point