Recursive Texture Mapping

Mapping a texture image onto a flat spatial surface and subsequently projecting the image into screen space (such as a computer display) by a central projection results in a fractional linear transformation. Hence the inverse texture map is fractional linear as well. The divisions involved in this inverse can be prohibitively expensive for real time applications. This paper presents a method that computes the texture coordinates by a simple recursive formula avoiding divisions. The texture map thus obtained only introduces very little distortion, is very efficient on mainstream hardware and is easy to implement. 1. Texture mapping basics In this section we will very briefly go over some texture mapping basics to get a feel for the kind of transformations involved. As in many applications we will restrict ourselves to texture mapping of triangles. Rendering a triangle involves three different spaces: texture space, object space and screen space. Texture space is 2-dimensional and defines the texture image to be mapped onto our given triangle. Object space is the 3-dimensional space in which the actual triangle is positioned. Screen space is 2-dimensional and can be thought of as embedded in object space. The mapping from object space onto screen space is a central projection as seen from some view point in object space. We will call the plane that passes through the view point and is parallel to screen space the base plane. Points on the base plane can not be mapped into screen space (in fact, projectively speaking, the base plane represents the “line at infinity” in screen space). We will assume that our triangle does not pass through the base space and is situated on the same side of it as the screen space. This means that the triangle is “visible” in its entirety from the view point. For convenience we also introduce triangle space as the hyperplane in object space that is spanned by the triangle. Let (T1, T2, T3) be the vertices in texture space of our triangular texture pattern. Let (P1, P2, P3) and (Q1, Q2, Q3) be the vertices of the triangle in object space and screen space respectively. Let (λ1, λ2, λ3) be affine coordinates in texture space, non-negative on the triangular texture, such that λi = 1 corresponds with vertex Ti. Similarly, let (μ1, μ2, μ3) and (η1, η2, η3) be affine coordinates in triangle space and screen space, non-negative on the (projected) triangle, such that μi = 1 and ηi = 1 correspond to Pi and Qi respectively. In these coordinates the texture map that maps Ti to Pi is now simply the identity (1.1) (x1, x2, x3) 7−→ (x1, x2, x3). The central projection of object space onto screen space also takes a rather simple form in these affine coordinates. Let z1, z2, z3 be the respective distances 1