Chapter Thirteen - Inferring Transforms: May-June 1999

Abstract

By picking the proper values for the coefficients a . j, one can fly the 2D texture around to an arbitrary position, orientation, and perspective projection on the screen. One can, in fact, generate the coefficients by a concatenation of 3D rotation, translation, scale, and perspective matrices. This chapter illustrates a more direct approach to finding a . j coefficients. It turns out that the 2D-to-2D mapping is completely specified if one gives four arbitrary points in screen space and the four arbitrary points in texture space. The only restriction is that in three of the input or output points any points should not be collinear. This method of transformation specification is useful, for example, in taking flat objects digitized in perspective and processing them into orthographic views. Paul Heckbert made a great leap by splitting the transformation into two separate matrices. He first used one matrix to map the input points to a canonical unit square (with vertices [0, 0], [1, 0], [1, 1], [0, 1]) and then mapped that square into the output points with another matrix. Each of these matrices will be individually easy to calculate than the complete transformation because the arithmetic is simple. In fact, the arithmetic turns out to be simplest for the second of these transformations.