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 Affine Transformation. 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 ?
j], assuring that G is also an affine transformation, as it is a composition of two affine transformations, thus getting,
Kannappan, "Method for 2-D affine transformation of images," US Patent No.
The initial step of the core algorithm resorts to SIFT features; in fact, they are quite well-known as being robust to rotation, scaling, and affine transformations and so they are well-suited for the forgeries recognition in near-duplicate images as has been debated in [5, 6].
Invariants of geometric moments with respect to affine transformations are generally called affine moment invariants.
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.
It is known as affine transformation as the transformation that results of the combination of translation, rotation and scale, defined by:
Apart from the standard affine transformations (scaling and translation), we use the transformations in Table I with the aid of which each admissible region can be transformed to a primitive region, as in Table II.
For local compositions, the precision with which classification can be undertaken is seen to satisfy the demands in semiotic theory for precision and rigor in analysis of the neutral level--his model deems objects at the neutral level "similar" if they are related by affine transformations.
Affine transformations can be appliedto any object--triangles, leaves, mountains, ferns, chimneys, clouds--or even the space in which an object sits.
1997), the connectivity properties of the Boolean model are invariant by affine transformations, which changes the shapes and the sizes, but does not introduce any new connection (this property was used to estimate the 3D connectivity number of a Boolean model of ellipsoids).
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.
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.