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Two intersecting lines. 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 set theory, a dichotomous relation R is such that either aRb, bRa, but not both. [1] A false dichotomy is an informal fallacy consisting of a supposed dichotomy which fails one or both of the conditions: it is not jointly exhaustive and/or not mutually exclusive. In its most common form, two entities are presented as if they are exhaustive ...
Consider a line L in the projective plane P 2: it has self-intersection number 1 since all other lines cross it once: one can push L off to L′, and L · L′ = 1 (for any choice) of L′, hence L · L = 1. In terms of intersection forms, we say the plane has one of type x 2 (there is only one class of lines, and they all intersect with each ...
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 (sometimes called a vertex ) or does not exist (if the lines are parallel ).
The intersection of A with any of B, C, D, or E is the empty set. In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their intersection is the point at
However, parallel (non-crossing) pairs of lines are less restricted in hyperbolic line arrangements than in the Euclidean plane: in particular, the relation of being parallel is an equivalence relation for Euclidean lines but not for hyperbolic lines. [51] The intersection graph of the lines in a hyperbolic arrangement can be an arbitrary ...
Let l 1 = [a 1, b 1, c 1] and l 2 = [a 2, b 2, c 2] be a pair of distinct lines. Then the intersection of lines l 1 and l 2 is point a P = (x 0, y 0, z 0) that is the simultaneous solution (up to a scalar factor) of the system of linear equations: a 1 x + b 1 y + c 1 z = 0 and a 2 x + b 2 y + c 2 z = 0. The solution of this system gives: x 0 ...
The converse is the Braikenridge–Maclaurin theorem, named for 18th-century British mathematicians William Braikenridge and Colin Maclaurin , which states that if the three intersection points of the three pairs of lines through opposite sides of a hexagon lie on a line, then the six vertices of the hexagon lie on a conic; the conic may be ...