# 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 ?
GC should be parameterized to establish one-to-one relation between points on GC of the original object and those on GC of its affine transformed version.
Then, a parametric point x([sigma]) = [[x([sigma]), y([sigma])].sup.T] on GC of the original object and a parametric point [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on GC of its affine transformed version satisfy the following equation:
Figure 4(b) shows the 10-shift curve of the affine transformed version Chinese character given in Figure 3(a).
Ben Yip (2005) developed face and eye rectification model using Affine transform. Affine transform matrix was used to rectify the face in video conferencing.
FACE AND EYE RECTIFICATION IN VIDEO CONFERENCE USING AFFINE TRANSFORM The National ICT Australia Limited, Australian Technology Park, Sydney, NSW, Australia The school of Information Technologies, University of Sydney, NSW, Australia y Moni., pp: 3-6.
Wu, "Detecting copy-move forgery under affine transforms for image forensics," Computers & Electrical Engineering, 2013.
3) Perform an inverse affine transform using equations (6) and (7) for the estimated source by Step 2 as (x',y') and the source location on the original coordinate system is determined as (x,y).
Here, [H.sub.ij] represents the affine transform matrix from node i to node j and is calculated by RANSAC algorithm.
When matching reference image (or moving image) with a certain node [n.sub.0], an estimated affine transform is calculated as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

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