OpenGL Programming Guide

Chapter 9Texture Mapping

Assigning Texture Coordinates

Computing Appropriate Texture Coordinates

Two-dimensional textures are square or rectangular images that are typically mapped to the polygons that make up a polygonal model. In the simplest case, you're mapping a rectangular texture onto a model that's also rectangular - for example, your texture is a scanned image of a brick wall, and your rectangle is to represent a brick wall of a building. Suppose the brick wall is square, and the texture is square, and you want to map the whole texture to the whole wall. The texture coordinates of the texture square are (0, 0), (1, 0), (1, 1), and (0, 1) in counterclockwise order. When you're drawing the wall, just give those four coordinate sets as the texture coordinates as you specify the wall's vertices in counterclockwise order.

Now suppose that the wall is two-thirds as high as it is wide, that the texture is again square. To avoid distorting the texture, you need to map the wall to a portion of the texture map so that the aspect ratio of the texture is preserved. Suppose that you decide to use the lower two-thirds of the texture map to texture the wall. In this case, use texture coordinates of (0,0), (1,0), (1,2/3), and (0,2/3) for the texture coordinates as the wall vertices are traversed in a counterclockwise order.

As a slightly more complicated example, suppose you'd like to display a tin can with a label wrapped around it on the screen. To obtain the texture, you purchase a can, remove the label, and scan it in. Suppose the label is 4 units tall and 12 units around, which yields an aspect ratio of 3 to 1. Since textures must have aspect ratios of 2n to 1, you can either simply not use the top third of the texture, or you can cut and paste the texture until it has the necessary aspect ratio. Let's say you decide to not use the top third. Now suppose the tin can is a cylinder approximated by thirty polygons of length 4 units (the height of the can) and width 12/30 (1/30 of the circumference of the can). You can use the following texture coordinates for each of the thirty approximating rectangles:

1: (0, 0), (1/30, 0), (1/30, 2/3), (0, 2/3)

2: (1/30, 0), (2/30, 0), (2/30, 2/3), (1/30, 2/3)

3: (2/30, 0), (3/30, 0), (3/30, 2/3), (2/30, 2/3)

. . .

30: (29/30, 0), (1, 0), (1, 2/3), (29/30, 2/3)

Only a few curved surfaces such as cones and cylinders can be mapped to a flat surface without geodesic distortion. Any other shape requires some distortion. In general, the higher the curvature of the surface, the more distortion of the texture is required.

If you don't care about texture distortion, it's often quite easy to find a reasonable mapping. For example, consider a sphere whose surface coordinates are given by (cos &thgr; cos &phgr; , cos &thgr; sin &phgr; , sin &thgr; ), where 0 ≤ &thgr; ≤ 2 &pgr; , and 0 ≤ &phgr; ≤ &pgr; . The &thgr; - &phgr; rectangle can be mapped directly to a rectangular texture map, but the closer you get to the poles, the more distorted the texture is. The entire top edge of the texture map is mapped to the north pole, and the entire bottom edge to the south pole. For other surfaces, such as that of a torus (doughnut) with a large hole, the natural surface coordinates map to the texture coordinates in a way that produces only a little distortion, so it might be suitable for many applications. Figure 9-5 shows two tori, one with a small hole (and therefore a lot of distortion near the center) and one with a large hole (and only a little distortion).

Figure 9-5 : Texture-Map Distortion

If you're texturing spline surfaces generated with evaluators (see Chapter 11 ), the u and v parameters for the surface can sometimes be used as texture coordinates. In general, however, there's a large artistic component to successfully mapping textures to polygonal approximations of curved surfaces.

OpenGL Programming Guide