# 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 ?
In this section, a new objective function is constructed, which minimizes an error of displacement estimation using a homogeneous affine transformation between pre- and post-deformation.
A solution is demonstrated which includes automated extraction of road intersections from high resolution aerial or satellite imagery (around 1 meter or finer where a road is no longer a thin line and road intersections are identifiable), automated matching of these with the equivalent intersections in a very large vector road map under both four-parameter similarity and six-parameter affine transformations, and a transformation verification and globalization procedure that handles error points and ensures optimization of the match between the image and the reference map.
The median of AED distances in pixels between automatically and manually derived landmarks positions for the supervising model (method M3) and the two-level optimization (method M4), given as a function of the number of PCA components of the affine transformation used in the statistical connection model.
and an affine transformation, for instance [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Each distorted, shrunken copy is defined by a particular affine transformation --a "contractive map,' as it's called-- of the whole leaf.
The term affine transformation comes from the Latin word affinis, meaning connected with; it describes a function that maps straight lines to straight lines (Wade and Sommer 2006).
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of Technology, Sydney) describes how to write a ray tracer step-by-step, starting with design and programming and some essential mathematics and theoretical foundations, moving to such details as antialiasing, sampling techniques, mapping samples to a disk or a hemisphere, perspective viewing and developing a practical viewing system, nonlinear projections, stereoscopy, lights and materials, specular reflection, shadows, ambient occlusion, area lights, ray-object interconnections, affine transformations, transforming objects, regular grids, triangle meshes, mirror reflections, global illumination, simple and realistic transparency, texture mapping, procedural textures and noise-based textures.
The invariance of the percolation threshold with respect to affine transformations in the common direction of the axis of cylinders is approximately satisfied on simulations.
The main features of IFS models are their simplicity and mathematical soundness: An IFS consists of a set of contractive affine transformations, which express a unique image (the attractor) in terms of selfsimilarities in a simple geometric way.
Subsequent chapters feature coverage of linear transformations from Rn to Rm, the geometry of linear and affine transformations, least squares fits and pseudoinverses, and eigenvalues and eigenvectors.
They supplement their text with exercises for classroom and self-study as they cover random vectors, including mathematical expectations and applications, Gaussian vectors, including affine transformations, discrete time processes, including an introduction to digital filtering, estimation, the Wiener filter, including its evaluation, adaptive filtering through an algorithm of the gradient and the LMS, and the Kalman filter.

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