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  2. Conformal linear transformation - Wikipedia

    en.wikipedia.org/.../Conformal_linear_transformation

    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 ...

  3. Homothety - Wikipedia

    en.wikipedia.org/wiki/Homothety

    Together with the translations, all homotheties of an affine (or Euclidean) space form a group, the group of dilations or homothety-translations. These are precisely the affine transformations with the property that the image of every line g is a line parallel to g .

  4. Beckman–Quarles theorem - Wikipedia

    en.wikipedia.org/wiki/Beckman–Quarles_theorem

    In geometry, the Beckman–Quarles theorem states that if a transformation of the Euclidean plane or a higher-dimensional Euclidean space preserves unit distances, then it preserves all Euclidean distances. Equivalently, every homomorphism from the unit distance graph of the plane to itself must be an isometry of the plane. The theorem is named ...

  5. Euclidean plane isometry - Wikipedia

    en.wikipedia.org/wiki/Euclidean_plane_isometry

    Beckman–Quarles theorem, a characterization of isometries as the transformations that preserve unit distances; Congruence (geometry) Coordinate rotations and reflections; Hjelmslev's theorem, the statement that the midpoints of corresponding pairs of points in an isometry of lines are collinear

  6. Affine transformation - Wikipedia

    en.wikipedia.org/wiki/Affine_transformation

    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 ...

  7. Möbius transformation - Wikipedia

    en.wikipedia.org/wiki/Möbius_transformation

    These transformations preserve angles, map every straight line to a line or circle, and map every circle to a line or circle. The Möbius transformations are the projective transformations of the complex projective line. They form a group called the Möbius group, which is the projective linear group PGL(2, C).

  8. Conformal group - Wikipedia

    en.wikipedia.org/wiki/Conformal_group

    In mathematics, the conformal group of an inner product space is the group of transformations from the space to itself that preserve angles. More formally, it is the group of transformations that preserve the conformal geometry of the space. Several specific conformal groups are particularly important: The conformal orthogonal group.

  9. Conformal Killing vector field - Wikipedia

    en.wikipedia.org/wiki/Conformal_Killing_vector_field

    In conformal geometry, a conformal Killing vector field on a manifold of dimension n with (pseudo) Riemannian metric (also called a conformal Killing vector, CKV, or conformal colineation), is a vector field whose (locally defined) flow defines conformal transformations, that is, preserve up to scale and preserve the conformal structure.