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Properties of this ternary operation have been used to define planar ternary rings in the foundations of projective geometry. In the Euclidean plane with points a, b, c referred to an origin, the ternary operation [,,] = + has been used to define free vectors. [2]
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 ...
In computational geometry, a CC system or counterclockwise system is a ternary relation pqr introduced by Donald Knuth to model the clockwise ordering of triples of points in general position in the Euclidean plane. [1]
In mathematics, a Euclidean plane is a Euclidean space of dimension two, denoted or . It is a geometric space in which two real numbers are required to determine the position of each point . It is an affine space , which includes in particular the concept of parallel lines .
Plane equation in normal form. In Euclidean geometry, a plane is a flat two-dimensional surface that extends indefinitely. Euclidean planes often arise as subspaces of three-dimensional space. A prototypical example is one of a room's walls, infinitely extended and assumed infinitesimal thin.
The archetypical example is the real projective plane, also known as the extended Euclidean plane. [4] 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.
If this property holds in the affine plane defined by a ternary ring, then there is an equivalence relation between "vectors" defined by pairs of points from the plane. [14] Furthermore, the vectors form an abelian group under addition; the ternary ring is linear and satisfies right distributivity: (+) = +.
The envelope of the tangent planes to M along a curve c is a surface with vanishing Gaussian curvature, which by Minding's theorem, must be locally isometric to the Euclidean plane. This identification allows parallel transport to be defined, because in the Euclidean plane all tangent planes are identified with the space itself.