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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 ...
Another important family of examples are the linear maps centred at an origin: given a point and a linear map , one may define an affine map ,: by , = + for every in . After making a choice of origin b {\displaystyle b} , any affine map may be written uniquely as a combination of a translation and a linear map centred at b {\displaystyle b} .
A linear map with viewed as a one ... For real numbers, the map + is not linear (but is an affine transformation). If is a real matrix, then defines a linear map from ...
In projective geometry, affine space means the complement of a hyperplane at infinity in a projective space. Affine space can also be viewed as a vector space whose operations are limited to those linear combinations whose coefficients sum to one, for example 2x − y, x − y + z, (x + y + z)/3, ix + (1 − i)y, etc.
A constant function is also considered linear in this context, as it is a polynomial of degree zero or is the zero polynomial. Its graph, when there is only one variable, is a horizontal line. In this context, a function that is also a linear map (the other meaning) may be referred to as a homogeneous linear function or a linear form.
The subgroup of the special affine group consisting of those transformations whose linear part has determinant 1 is the group of orientation- and volume-preserving maps. Algebraically, this group is a semidirect product S L ( V ) ⋉ V {\displaystyle SL(V)\ltimes V} of the special linear group of V {\displaystyle V} with the translations.
See Affine space § Affine combinations and barycenter for the definition in this case. This concept is fundamental in Euclidean geometry and affine geometry , because the set of all affine combinations of a set of points forms the smallest affine space containing the points, exactly as the linear combinations of a set of vectors form their ...
Additionally, they try to do linear interpolation along a line of pixels to simplify the set-up (compared to 2d affine interpolation) and thus again the overhead (also affine texture-mapping does not fit into the low number of registers of the x86 CPU; the 68000 or any RISC is much more suited).