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Collinearity of points whose coordinates are given [ edit ] In coordinate geometry , in n -dimensional space, a set of three or more distinct points are collinear if and only if, the matrix of the coordinates of these vectors is of rank 1 or less.
The collinearity equations are a set of two equations, ... Denote the coordinates of the point P on the object by ,, , the coordinates of the image ...
This is different from the behavior in higher dimensions, and thus one gives a more restrictive definition, specified so that the fundamental theorem of projective geometry holds. In this definition, when V has dimension two, a collineation from PG(V) to PG(W) is a map α : D(V) → D(W), such that:
with the notation : defined to mean the signed ratio of the displacement from W to X to the displacement from Y to Z. For colinear displacements this is a dimensionless quantity. If the displacements themselves are taken to be signed real numbers, then the cross ratio between points can be written
Let X be an affine space over a field k, and V be its associated vector space. An affine transformation is a bijection f from X onto itself that is an affine map; this means that a linear map g from V to V is well defined by the equation () = (); here, as usual, the subtraction of two points denotes the free vector from the second point to the first one, and "well-defined" means that ...
Trilinear coordinates are an example of homogeneous coordinates. The ratio x : y is the ratio of the perpendicular distances from the point to the sides (extended if necessary) opposite vertices A and B respectively; the ratio y : z is the ratio of the perpendicular distances from the point to the sidelines opposite vertices B and C ...
a coordinate line, a linear coordinate dimension; In the context of determining parallelism in Euclidean geometry, a transversal is a line that intersects two other lines that may or not be parallel to each other. For more general algebraic curves, lines could also be: i-secant lines, meeting the curve in i points counted without multiplicity, or
The problem they apply to involves placing the vertices of a given graph at integer coordinates in the plane, and drawing the edges of the graph as straight line segments. For certain graphs, such as the utility graph , crossings between pairs of edges are unavoidable, but one should still avoid placements that cause a vertex to lie on an edge ...