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2. 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 ...

3. Affine group - Wikipedia

   en.wikipedia.org/wiki/Affine_group

   In mathematics, the affine group or general affine group of any affine space is the group of all invertible affine transformations from the space into itself. In the case of a Euclidean space (where the associated field of scalars is the real numbers), the affine group consists of those functions from the space to itself such that the image of every line is a line.

4. Affine shape adaptation - Wikipedia

   en.wikipedia.org/wiki/Affine_shape_adaptation

   An important consequence of this study is that if we can find an affine transformation such that is a constant times the unit matrix, then we obtain a fixed-point that is invariant to affine transformations (Lindeberg 1994, section 15.4; Lindeberg & Garding 1997). For the purpose of practical implementation, this property can often be reached ...

5. Homothety - Wikipedia

   en.wikipedia.org/wiki/Homothety

   These are precisely the affine transformations with the property that the image of every line g is a line parallel to g. In projective geometry, a homothetic transformation is a similarity transformation (i.e., fixes a given elliptic involution) that leaves the line at infinity pointwise invariant. [2]

6. Affine curvature - Wikipedia

   en.wikipedia.org/wiki/Affine_curvature

   The special affine curvature of an immersed curve is the only (local) invariant of the curve in the following sense: If two curves have the same special affine curvature at every point, then one curve is obtained from the other by means of a special affine transformation. In fact, a slightly stronger statement holds:

7. Affine involution - Wikipedia

   en.wikipedia.org/wiki/Affine_involution

   Affine involutions can be categorized by the dimension of the affine space of fixed points; this corresponds to the number of values 1 on the diagonal of the similar matrix D (see above), i.e., the dimension of the eigenspace for eigenvalue 1. The affine involutions in 3D are: the identity; the oblique reflection in respect to a plane