Affine and Projective Transformations
In addition to isometries, there are two kinds of mappings that preserve lines: affine (Section 3.1 ) and projective (Section 3.2 ) transformations. Affine transformations f of \({\mathbb{R}}^{n}\) have the following property: If l is a line then f(l) is
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Affine and Projective Transformations
In addition to isometries, there are two kinds of mappings that preserve lines: affine (Section 3.1) and projective (Section 3.2) transformations. Affine transformations f of Rn have the following property: If l is a line then f (l) is also a line, and if l k then f (l) f (k). A line in Rn means a set of the form {r0 + r : r ∈ W }, where r0 ∈ Rn and W ⊂ Rn is a one-dimensional subspace. Projective transformations f of Rn map lines to lines, preserving the cross-ratio of four points. We also use homogeneous coordinates x = (x1 : . . . : xn+1) in Rn+1 . Section 3.3 describes transformation matrices in homogeneous coordinates.
3.1 Affine Transformations An affine transformation f : Rn → Rm is a linear mapping followed by a translation, i.e., f (v) = A(v) + b, where b ∈ Rm and A : Rn → Rm is an invertible linear mapping. The set Affn (R) of affine transformations of Rn is a group under the operation of composition.
3.1.1 Affine transformations in two and three dimensions An affine transformation of the plane, f : R2 → R2 , has the form f (x, y) = (a11 x + a12y + b1 , a21 x + a22 y + b2) R V. Rovenski, Modeling of Curves and Surfaces with MATLAB , Springer Undergraduate Texts in Mathematics and Technology 7, DOI 10.1007/978-0-387-71278-9 3, c Springer Science+Business Media, LLC 2010
(3.1) 135
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3 Affine and Projective Transformations
for some real numbers ai j , b j satisfying Δ = a11 a22 − a12 a21 = 0. a11 a21 2 In matrix form we write f (x) = xA + b (x ∈ R ), where A = is an inverta12 a22 ible matrix, and b = (b1 , b2 ) ∈ R2 . One may use (3.1) to verify the properties of affine transformations: (1) mapping lines to lines; (2) mapping parallel lines to parallel lines; (3) preserving ratios of lengths along a given line. − → −→ If A, B, C, D ∈ R2 are four collinear points (with C = D), the ratio AB/CD is − → −→ well defined; this is a scalar λ ∈ R such that AB = λ CD. The property (3) means . − → −→ −−−−−−→ −−−−−−→ that AB/CD = f (A) f (B) f (C) f (D). Fundamental theorem of affine geometry. (For simplicity, consider the two-dimensional case.) Let A, B, C and A , B , C be two sets of three non-collinear points in R2 . Then there is a unique affine transformation f that maps A, B, C to A , B , C , respectively. Next, we consider particular cases of affine transformations. A scaling about the origin is a transformation that maps a point P = (x, y) to a point P = (x , y ) by multiplying the x and y coordinates by positive constant scaling factors λx 0 λx and λy , respectively, to give x = λx x and y = λy y. The matrix S(λx , λy ) = 0 λy is called the scaling transformation matrix. If λx = 1 and λy = 1, we have a horizontal scaling Sx (λx ), while if λy = 1 and λx = 1, we have a vertical scaling Sy (λy ): x = λx x, x = x, and y =y y = λy y. A scaling transformation is uniform (another name is a dilation about O) whenever λx = λy = λ . A scaling factor λ is said to be an expansion if λ > 1, and a compression if λ < 1. If the constant
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