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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 particular it is important to assure that for two given line segments, a new line segment can be constructed, such that its length equals the product of lengths of the other two. Similarly one needs to be able to construct, for a line segment of length , a new line segment of length . The intercept theorem can be used to show that for both ...
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 ...
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 ).
In geometry, a line segment is a part of a straight line that is bounded by two distinct endpoints (its extreme points), and contains every point on the line that is between its endpoints. It is a special case of an arc , with zero curvature .
In algebraic geometry, standard monomial theory describes the sections of a line bundle over a generalized flag variety or Schubert variety of a reductive algebraic group by giving an explicit basis of elements called standard monomials. Many of the results have been extended to Kac–Moody algebras and their groups.
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
For example, the first Napoleon point is the point of concurrency of the three lines each from a vertex to the centroid of the equilateral triangle drawn on the exterior of the opposite side from the vertex. A generalization of this notion is the Jacobi point. The de Longchamps point is the point of concurrence of several lines with the Euler line.