Homogeneous Coordinates and Computer Graphics

Arya MirΛογισμικό & κατασκευή λογ/κού

9 Σεπ 2011 (πριν από 5 χρόνια και 2 μήνες)

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The relationship between Cartesian coordinates and Euclidean geometry is well known. The theorems from Euclidean geometry don’t mention anything about coordinates, but when you need to apply those theorems to a physical problem, you need to calculate lengths, angles, et cetera, or to do geometric proofs using analytic geometry. Homogeneous coordinates and projective geometry bear exactly the same relationship. Homogeneous coordinates provide a method for doing calculations and proving theorems in projective geometry, especially when it is used in practical applications. Although projective geometry is a perfectly good area of “pure mathematics”, it is also quite useful in certain real-world applications. The one with which the author is most familiar is in the area of computer graphics. Since it is almost always easier to understand mathematics when there are concrete examples available, we’ll use computer graphics in this document as a source for almost all the examples. The prerequisites for the material contained herein include matrix algebra (how to multiply, add, and invert matrices, and how to multiply vectors by matrices to obtain other vectors), a bit of vector algebra, some trigonometry, and an understanding of Euclidean geometry.

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