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k = −1 corresponds to a point reflection at point S Homothety of a pyramid. In mathematics, a homothety (or homothecy, or homogeneous dilation) is a transformation of an affine space determined by a point S called its center and a nonzero number k called its ratio, which sends point X to a point X ′ by the rule, [1]
Conformal linear transformations come in two types, proper transformations preserve the orientation of the space whereas improper transformations reverse it. As linear transformations, conformal linear transformations are representable by matrices once the vector space has been given a basis , composing with each-other and transforming vectors ...
The point z ∗ is conjugate to z when C is the circle of a radius r, centered about z 0. This can be explicitly given as = ¯ +. Since Möbius transformations preserve generalized circles and cross-ratios, they also preserve the conjugation.
These transformations are exactly those which preserve a kind of squared distance between oriented circles called their Darboux product. The direct Laguerre transformations are defined as the subgroup , + (,). In 2 dimensions, the direct Laguerre transformations can be represented by 2×2 dual number matrices.
The conformal property may be described in terms of the Jacobian derivative matrix of a coordinate transformation. The transformation is conformal whenever the Jacobian at each point is a positive scalar times a rotation matrix (orthogonal with determinant one). Some authors define conformality to include orientation-reversing mappings whose ...
Conformal transformations preserve angles, and are, in the first order, similarities. Equiareal transformations, preserve areas in the planar case or volumes in the three dimensional case. [9] and are, in the first order, affine transformations of determinant 1. Homeomorphisms (bicontinuous transformations) preserve the neighborhoods of points.
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 ...
It follows from this that any transformation of the plane that preserves the unit distances in must also preserve the distance between and . [ 16 ] [ 17 ] [ 18 ] A. D. Alexandrov asked which metric spaces have the same property, that unit-distance-preserving mappings are isometries, [ 19 ] and following this question several authors have ...