Intersection (geometry)

In geometry, an intersection is a point, line, or curve common to two or more objects (such as lines, curves, planes, and surfaces). The simplest case in Euclidean geometry is the line–line intersection between two distinct lines, which either is one point or does not exist (if the lines are parallel). Other types of geometric intersection include: Line–plane intersection Line–sphere intersection Intersection of a polyhedron with a line Line segment intersection Intersection curve Determination of the intersection of flats – linear geometric objects embedded in a higher-dimensional space – is a simple task of linear algebra, namely the solution of 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 Newton iteration. Intersection problems between a line and a conic section (circle, ellipse, parabola, etc.) or a quadric (sphere, cylinder, hyperboloid, etc.) lead to quadratic equations that can be easily solved. Intersections between quadrics lead to quartic equations that can be solved algebraically. Line–line intersection For the determination of the intersection point of two non-parallel lines one gets, from Cramer's rule or by substituting out a variable, the coordinates of the intersection point : (If the lines are parallel and these formulas cannot be used because they involve dividing by 0.) Multiple line segment intersection and Line–line_intersection#Given_two_points_on_each_line_segment For two non-parallel line segments and there is not necessarily an intersection point (see diagram), because the intersection point of the corresponding lines need not to be contained in the line segments. In order to check the situation one uses parametric representations of the lines: The line segments intersect only in a common point of the corresponding lines if the corresponding parameters fulfill the condition . The parameters are the solution of the linear system It can be solved for s and t using Cramer's rule (see above).