齐次坐标与计算机图形.pdfVIP

Homogeneous Coordinates and Computer Graphics Tom Davis tomrdavis@ /mathcircles November 20, 2001 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 co- ordinates 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. 1 Computer Graphics Problems We’ll begin the study of homogeneous coordinates by describing a set of problems from three-dimensional computer graphics that at first seem to have unrelated solutions. We will then show that with certain “tricks”, all of them can be solved in the same way. Finally, we will show that this “same way” is in fact just a recasting of