"The principal difference is between a single rotation and two different orthogonal matrices. This difference causes another, less important, difference. Because the SVD has different singular vectors on the two sides, there is no need for negative Singular values: we can always flip the sign of a si"
"However, this type of transformation, in which one of the coordinates of the input vector appears in the denominator, can’t be achieved using affine transformations. We can allow for division with a simple generalization of the mechanism of homogeneous coordinates that we have been using for affine "
"Managing coordinate systems is one of the core tasks of almost any graphics program; key to this is managing orthonormal bases."
"The advantages of parallel projection are also its limitations. In our everyday experience (and even more so in photographs) objects look smaller as they get farther away, and as a result parallel lines receding into the distance do not ap- pear parallel. This is because eyes and cameras don’t colle"
"1.Rotate v_1 and v_2 to the x- and y-axes (the transform by R^T). 2.Scale in x and y by (λ_1,λ_2)(the transform by S). 3.Rotate the x- and y-axes back to v_1 and v_2 (the transform by R). Looking at the effect of these three transforms together, we can see that they have the effect of a nonuniform s"
"If you like to count dimensions: a symmetric 2×2 matrix has 3° of freedom, and the eigenvalue decomposition rewrites them as a rotation angle and two scale factors."
"A very similar kind of decomposition can be done with non symmetric matrices as well: it's the singular value decomposition(SVD), also discussed in section 6.4.1. The difference is that the matrices on either side of the dialogue matrix are no longer the same: A=USV^T The two orthogonal matrices tha"
"In summary, every matrix can be decomposed via SVD into a rotation times a scale times another rotation. Only symmetric matrices can be decomposed via eigenvalue diagonalization into a rotation times a scale times the inverse-rotation, and such matrices are a simple scale in an arbitrary direction. "
作者简介
舍利,计算机图形学领域世界知名的学者,尤以光线跟踪方面的研究闻名世界,曾担任ACM Transactions on Graaphics和Jouranl of Graphics Tools副主编,多次担任sIGGRAPH程序委员会委员。他是犹他大学计算机科学系教授。在伊利诺伊大学厄巴纳·尚佩恩分校获得计算机科学博士学位。除本书之外,他还著有Realistic Ray Tracing。