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Effect of applying various 2D affine transformation matrices on a unit square. Note that the reflection matrices are special cases of the scaling matrix. Affine transformations on the 2D plane can be performed in three dimensions. Translation is done by shearing parallel to the xy plane, and rotation is performed around the z axis.
The similarity transformations form the subgroup where is a scalar times an orthogonal matrix. For example, if the affine transformation acts on the plane and if the determinant of is 1 or −1 then the transformation is an equiareal mapping. Such transformations form a subgroup called the equi-affine group. [13]
2D affine transformation matrix: Image title: Illustration of the effect of applying various 2D affine transformation matrices on a unit square by CMG Lee. Note that the reflection matrices are special cases of the scaling matrix. Width: 100%: Height: 100%
Thus every shear matrix has an inverse, and the inverse is simply a shear matrix with the shear element negated, representing a shear transformation in the opposite direction. In fact, this is part of an easily derived more general result: if S is a shear matrix with shear element λ, then S n is a shear matrix whose shear element is simply nλ.
Noting that any identity matrix is a rotation matrix, and that matrix multiplication is associative, we may summarize all these properties by saying that the n × n rotation matrices form a group, which for n > 2 is non-abelian, called a special orthogonal group, and denoted by SO(n), SO(n,R), SO n, or SO n (R), the group of n × n rotation ...
2D perspective transformation matrix: Image title: Comparison of the effects of applying 2D affine and perspective transformation matrices on a unit square by CMG Lee. In this example, a = 3, b = 4, c = 5, d = 6, e = 2, f = 4, g = 2 and h = 1. Width: 100%: Height: 100%
Formulas involving homogeneous coordinates are often simpler and more symmetric than their Cartesian counterparts. Homogeneous coordinates have a range of applications, including computer graphics and 3D computer vision, where they allow affine transformations and, in general, projective transformations to be easily represented by a matrix.
Since a translation is an affine transformation but not a linear transformation, homogeneous coordinates are normally used to represent the translation operator by a matrix and thus to make it linear. Thus we write the 3-dimensional vector w = (w x, w y, w z) using 4 homogeneous coordinates as w = (w x, w y, w z, 1). [1]