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In the Euclidean plane with points a, b, c referred to an origin, the ternary operation [,,] = + has been used to define free vectors. [2] Since ( abc ) = d implies b – a = c – d , the directed line segments b – a and c – d are equipollent and are associated with the same free vector.
The archetypical example is the real projective plane, also known as the extended Euclidean plane. [1] This example, in slightly different guises, is important in algebraic geometry , topology and projective geometry where it may be denoted variously by PG(2, R ) , RP 2 , or P 2 ( R ), among other notations.
The eight (±,±,±) coordinates of the cube vertices are used to denote them. The horizontal plane shows the four quadrants between x- and y-axis. (Vertex numbers are little-endian balanced ternary.) An octant in solid geometry is one of the eight divisions of a Euclidean three-dimensional coordinate system defined
A planar ternary ring (PTR) or ternary field is special type of ternary system used by Marshall Hall [1] to construct projective planes by means of coordinates. A planar ternary ring is not a ring in the traditional sense, but any field gives a planar ternary ring where the operation T {\displaystyle T} is defined by T ( a , b , c ) = a b + c ...
The affine coordinates in a Desarguesian plane for the points designated to be the points at infinity (in this example: C, E and G) can be defined in several other ways. In standard notation, a finite projective geometry is written PG(a, b) where: a is the projective (or geometric) dimension, and
The most basic example is the flat Euclidean plane, an idealization of a flat surface in physical space such as a sheet of paper or a chalkboard. On the Euclidean plane, any two points can be joined by a unique straight line along which the distance can be measured.
The term triangular coordinates may refer to any of at least three related systems of coordinates in the Euclidean plane: a special case of barycentric coordinates for a triangle, in which case it is known as a ternary plot or areal coordinates, among other names
If no ambient space is mentioned then the Euclidean plane is assumed. The Fano plane (example 1 above) is not realizable since it needs at least one curve. The Möbius–Kantor configuration (example 4 above) is not realizable in the Euclidean plane, but it is realizable in the complex plane. [7] On the other hand, examples 2 and 5 above are ...