affine transformation

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affine transformation

[ə′fīn ‚tranz·fər′mā·shən]
(mathematics)
A function on a linear space to itself, which is the sum of a linear transformation and a fixed vector.

Affine Transformation

a pointwise mutually single-valued mapping of a plane (space) onto itself in which straight lines are transformed into straight lines. If a Cartesian coordinate system is given in a plane, then any affine transformation of this plane can be defined by means of a so-called nonsingu-lar linear transformation of the coordinates x and y of the points of this plane. Such a transformation is given by the formulas x’ = ax + by + p and y’ = cx + dy + q with the additional requirement that . Analogously, any affine transformation of a space can be defined by means of nonsingular linear transformations of the coordinates of points in space. The set of all affine transformations of a plane (space) into itself forms a group of affine transformations. This denotes, in particular, that the successive execution of two affine transformations is equivalent to some single affine transformation.

Examples of affine transformations are the orthogonal transformation—a motion of a plane or space or motion with a reflection; the transformation of similitude; and uniform “compression.” A uniform “compression” with coefficient k of the plane π toward a straight line a located in it is a transformation in which the points of a remain stationary and every point M of the plane π which does not lie on a is displayed along a ray passing through M perpendicularly to a to a point M’ such that the ratio of the distances from M and M’ to a is equal to k. Analogously, one defines a uniform “compression” of space to a plane. Each affine transformation of the plane can be obtained by performing a certain orthogonal transformation and a successive “compression” on some two perpendicular lines. Any affine transformation of space can be accomplished by means of a certain orthogonal transformation and successive “compressions” on some three mutually perpendicular lines. In an affine transformation, parallel lines and planes are transformed into parallel lines and planes. The properties of the affine transformation are widely used in various branches of mathematics, mechanics, and theoretical physics. Thus, in geometry the affine transformation is used for the so-called affine classification of figures. In mechanics, it is used in the study of small deformations of continuous media; in such deformations, small elements of the medium in the first approximation undergo affine transformations.

REFERENCES

Muskhelishvili, N. I. Kurs analiticheskoi geometrii, 4th ed. Moscow, 1967.
Aleksandrov, P. S. Lektsii po analiticheskoi geometrii. Moscow, 1968.
Efimov, N. V. Vysshaia geometriia, 4th ed. Moscow, 1961.
E. G. POZNIAK

affine transformation

(mathematics)
A linear transformation followed by a translation. Given a matrix M and a vector v,

A(x) = Mx + v

is a typical affine transformation.
References in periodicals archive ?
If C affinely uniformly embeds into a reflexive space Y, then C has the fixed point property for continuous affine mappings.
It is easy to see that [mathematical expression not reproducible] has the fixed point property for continuous affine mappings by Schauder-Tychonoff theorem  and [mathematical expression not reproducible] does not have the fixed point property for continuous affine mappings by Theorems 3.2 in .
On the other hand, optical scaling occurs when the shape undergoes unbalanced homothety (dilation) or some transformations which are not affine mappings.
To enhance the ability to detect interval changes in subtle lung lesions by using a subtraction image, we used MI lung mask affine mapping combined with CC lung tissue greedy SyN mapping to accurately register temporal serial chest CT images.
The previous characterization allows us to obtain, in section 5, the property of approximate contraction for affine mappings. The link with errors arising in floating point arithmetic is developped in section 6 in the case of rounding and chopping and in the context of parallel asynchronous linear iterations.
Let [X.sub.1] be the dimensional extension from X by an affine mapping [X.sub.1] = XQ + [??][b.sup.T].
Then there exists an affine mapping that maps [X.sub.1] to [X.sub.2].
He calls upon abstract algebra to demonstrate that those local-composition symmetrics and automorphisms that are affine mappings always possess a group structure.
Some IFSs of affine mappings are not necessarily contractions with respect to the Euclidean metric but are essential contractions (see ).

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