Shear mapping
In
An example is the
Shear mappings must not be confused with
The same definition is used in
In the general n-dimensional
Definition
Horizontal and vertical shear of the plane
In the plane , a horizontal shear (or shear parallel to the x-axis) is a function that takes a generic point with coordinates to the point ; where m is a fixed parameter, called the shear factor.
The effect of this mapping is to displace every point horizontally by an amount proportionally to its y-coordinate. Any point above the x-axis is displaced to the right (increasing x) if m > 0, and to the left if m < 0. Points below the x-axis move in the opposite direction, while points on the axis stay fixed.
Straight lines parallel to the x-axis remain where they are, while all other lines are turned (by various angles) about the point where they cross the x-axis. Vertical lines, in particular, become oblique lines with slope Therefore, the shear factor m is the
If the coordinates of a point are written as a
A vertical shear (or shear parallel to the y-axis) of lines is similar, except that the roles of x and y are swapped. It corresponds to multiplying the coordinate vector by the
The vertical shear displaces points to the right of the y-axis up or down, depending on the sign of m. It leaves vertical lines invariant, but tilts all other lines about the point where they meet the y-axis. Horizontal lines, in particular, get tilted by the shear angle to become lines with slope m.
Composition
Two or more shear transformations can be combined.
If two shear matrices are and
then their composition matrix is
In particular, if , we have
which is a
Higher dimensions
A typical shear matrix is of the form
This matrix shears parallel to the x axis in the direction of the fourth dimension of the underlying vector space.
A shear parallel to the x axis results in and . In matrix form:
Similarly, a shear parallel to the y axis has and . In matrix form:
In 3D space this matrix shear the YZ plane into the diagonal plane passing through these 3 points:
The
Properties
If S is an n × n shear matrix, then:
- S has rank n and therefore is invertible
- 1 is the only S = n
- the eigenspaceof S (associated with the eigenvalue 1) has n − 1 dimensions.
- S is defective
- S is asymmetric
- S may be made into a block matrix by at most 1 column interchange and 1 row interchange operation
- the volume, or any higher order interior capacity of a polytopeis invariant under the shear transformation of the polytope's vertices.
General shear mappings
For a vector space V and subspace W, a shear fixing W translates all vectors in a direction parallel to W.
To be more precise, if V is the
correspondingly, the typical shear L fixing W is
where M is a linear mapping from W′ into W. Therefore in block matrix terms L can be represented as
Applications
The following applications of shear mapping were noted by William Kingdon Clifford:
- "A succession of shears will enable us to reduce any figure bounded by straight lines to a triangle of equal area."
- "... we may shear any triangle into a right-angled triangle, and this will not alter its area. Thus the area of any triangle is half the area of the rectangle on the same base and with height equal to the perpendicular on the base from the opposite angle."[2]
The area-preserving property of a shear mapping can be used for results involving area. For instance, the Pythagorean theorem has been illustrated with shear mapping[3] as well as the related geometric mean theorem.
Shear matrices are often used in computer graphics.[4][5][6]
An algorithm due to Alan W. Paeth uses a sequence of three shear mappings (horizontal, vertical, then horizontal again) to rotate a digital image by an arbitrary angle. The algorithm is very simple to implement, and very efficient, since each step processes only one column or one row of pixels at a time.[7]
In typography, normal text transformed by a shear mapping results in oblique type.
In pre-Einsteinian
See also
References
- ^ Definition according to Weisstein, Eric W. Shear From MathWorld − A Wolfram Web Resource
- ^ William Kingdon Clifford (1885) Common Sense and the Exact Sciences, page 113
- ^ Hohenwarter, M Pythagorean theorem by shear mapping; made using GeoGebra. Drag the sliders to observe the shears
- ^ Foley et al. (1991, pp. 207–208, 216–217)
- ^ Geometric Tools for Computer Graphics, Philip J. Schneider and David H. Eberly, pp. 154-157
- ^ Computer Graphics, Apueva A. Desai, pp. 162-164
- ^ A.W. Paeth (1986), A Fast Algorithm for General Raster Rotation. Vision Interface (VI1986) pp 077-081.
Bibliography
- Foley, James D.; van Dam, Andries; Feiner, Steven K.; Hughes, John F. (1991), Computer Graphics: Principles and Practice (2nd ed.), Reading: ISBN 0-201-12110-7