Abstract
A geometry (affine, Euclidean, conformal, projective, or any kind) is characterized by certain operators that act on the objects of the geometry. These objects can range from geometrical entities such as a triangle to properties such as length, so they may have various dimensionalities that are called grades. The operators in the geometry change these objects in a covariant manner. A convenient algebraic system encodes the operators so deeply into its framework that such covariant identities hold trivially. Conversely, only elements that transform in this manner are considered as objects in the geometry and other elements do not have permanence of their defining properties. Geometric algebra offers a method to produce such an automatically covariant representation system. Finding a good representational space is required in order to use the structural capability for a given geometry. In this representational space, the symmetries should become isometries that are distance-preserving transformations. They are the orthogonal transformations in the representational space that can be represented by versors in the corresponding geometric algebra.
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