ch04oddslns - Angel: Interactive Computer Graphics, Fourth...

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Angel: Interactive Computer Graphics, Fourth Edition Chapter 4 Solutions 4.1 If the scaling matrix is uniform then RS = RS ( α, α, α )= α R = SR Consider R x ( θ ), if we multiply and use the standard trigonometric identities for the sine and cosine of the sum of two angles, we Fnd R x ( θ ) R x ( φ R x ( θ + φ ) By simply multiplying the matrices we Fnd T ( x 1 ,y 1 ,z 1 ) T ( x 2 2 2 T ( x 1 + x 2 1 + y 2 1 + z 2 ) 4.5 There are 12 degrees of freedom in the three–dimensional affine transformation. Consider a point p =[ x, y, z, 1] T that is transformed to p 0 x 0 y 0 0 , 1] T by the matrix M . Hence we have the relationship p 0 = Mp where M has 12 unknown coefficients but p and p 0 are known. Thus we have 3 equations in 12 unknowns (the fourth equation is simply the identity 1=1). If we have 4 such pairs of points we will have 12 equations in 12 unknowns which could be solved for the elements of M . Thus if we know how a quadrilateral is transformed we can determine the affine transformation. In two dimensions, there are 6 degrees of freedom in M but p and p 0 have only x and y components. Hence if we know 3 points both before and after transformation, we will have 6 equations in 6 unknowns and thus in two
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ch04oddslns - Angel: Interactive Computer Graphics, Fourth...

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