In geometry, an intersection between geometric objects (seen as sets of points) 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 (sometimes called a vertex) or empty (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. The notion of intersection from geometry has been exended to the status of an operation with sets, intersection (set theory), in works by Giuseppe Peano.
On a plane
Two lines
For the determination of the intersection point of two non-parallel lines
a 1 x + b 1 y = c 1 , a 2 x + b 2 y = c 2 {\displaystyle a_{1}x+b_{1}y=c_{1},\ a_{2}x+b_{2}y=c_{2}} one gets, from Cramer's rule or by substituting out a variable, the coordinates of the intersection point ( x s , y s ) {\displaystyle (x_{s},y_{s})} :
x s = c 1 b 2 − c 2 b 1 a 1 b 2 − a 2 b 1 , y s = a 1 c 2 − a 2 c 1 a 1 b 2 − a 2 b 1 . {\displaystyle x_{s}={\frac {c_{1}b_{2}-c_{2}b_{1}}{a_{1}b_{2}-a_{2}b_{1}}},\quad y_{s}={\frac {a_{1}c_{2}-a_{2}c_{1}}{a_{1}b_{2}-a_{2}b_{1}}}.\ }
(If a 1 b 2 − a 2 b 1 = 0 {\displaystyle a_{1}b_{2}-a_{2}b_{1}=0} the lines are parallel and these formulas cannot be used because they involve dividing by 0.)
Two line segments
For two non-parallel line segments ( x 1 , y 1 ) , ( x 2 , y 2 ) {\displaystyle (x_{1},y_{1}),(x_{2},y_{2})} and ( x 3 , y 3 ) , ( x 4 , y 4 ) {\displaystyle (x_{3},y_{3}),(x_{4},y_{4})} there is not necessarily an intersection point (see diagram), because the intersection point ( x 0 , y 0 ) {\displaystyle (x_{0},y_{0})} 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:
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