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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.
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]
Reflection. Reflections, or mirror isometries, denoted by F c,v, where c is a point in the plane and v is a unit vector in R 2.(F is for "flip".) have the effect of reflecting the point p in the line L that is perpendicular to v and that passes through c.
Every origin-fixing reflection or dilation is a conformal linear transformation, as is any composition of these basic transformations, including rotations and improper rotations and most generally similarity transformations. However, shear transformations and non-uniform scaling are not. Conformal linear transformations come in two types ...
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.
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 ...
Case 5 corresponds to a shear mapping combined with a dilation. Case 6 corresponds to similarities when the coordinate axes are perpendicular. The affine transformations without any fixed point belong to cases 1, 3, and 5. The transformations that do not preserve the orientation of the plane belong to cases 2 (with ab < 0) or 3 (with a < 0).
A conformal transformation on S is a projective linear transformation of P(R n+2) that leaves the quadric invariant. In a related construction, the quadric S is thought of as the celestial sphere at infinity of the null cone in the Minkowski space R n +1,1 , which is equipped with the quadratic form q as above.