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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 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 (red) of two disks (white and red with black boundaries). The circle (black) intersects the line (purple) in two points (red). The disk (yellow) intersects the line in the line segment between the two red points. The intersection of D and E is shown in grayish purple. The intersection of A with any of B, C, D, or E is the empty ...
For example, for any two distinct points, there is a unique line containing them, and any two distinct lines intersect at most at one point. [1]: 300 In two dimensions (i.e., the Euclidean plane), two lines that do not intersect are called parallel.
In analytic geometry, the intersection of a line and a plane in three-dimensional space can be the empty set, a point, or a line. It is the entire line if that line is embedded in the plane, and is the empty set if the line is parallel to the plane but outside it. Otherwise, the line cuts through the plane at a single point.
The computation of the intersection of two lines shows that the entire pencil of lines centered at a point is determined by any two of the lines that intersect at that point. It immediately follows that the algebraic condition for three lines, [a 1, b 1, c 1], [a 2, b 2, c 2], [a 3, b 3, c 3] to be concurrent is that the determinant,
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 ...
Lines A, B and C are concurrent in Y. In geometry, lines in a plane or higher-dimensional space are concurrent if they intersect at a single point.. The set of all lines through a point is called a pencil, and their common intersection is called the vertex of the pencil.