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FIGURE 32.8 Unit cell of body-centered cubic structure. (a) The unit cell has 1 8 atom at each of 8 corners and 1 atom at the geometric center of the cell, for a total of 2 atoms; (b) the relationship of the lattice parameter a and atomic radius r.

That s quite a bit of work to do for each vertex, especially with those square roots on the iPhone CPU. Fortunately, you can make some approximations again that will let you speed things up significantly. The key observation is that you could fake the reflection vector by simply using the normal vector at each vertex. The reflection wouldn t be physically accurate, but it would reflect something, which is good enough. That by itself isn t enough, though: if you use only the normal vector, the reflection wouldn t change as the camera moves around the object or the object changes position. To fix that, you need to rotate the vertex normal by the camera transform. Finally, you discard the z component of the normal and scale and bias the x and y components so they are between 0 and 1. The results are the texture coordinates corresponding to the environment mapping as seen from the correct viewpoint. This technique is sometimes referred to as normal environment mapping, because it uses the normals of the model instead of the true reflection vectors.

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the axis and the origin measured in terms of multiples or fractions of the unit cell lengths a, b, and c used in the determination The final steps in specifying the Miller indices are to reduce the three reciprocals to the lowest integers having the same ratio and then to enclose them in parentheses As is true with direction indices, the sequence of integers relates to the distances along the x, y, and z axes, respectively The following examples should make this procedure clear Figure 329a identifies the front face of the crystal with the Miller indices (100) This notation is arrived at as follows: The front face intercepts the x axis at one a distance, and it does not intercept the y and z axes (or it intercepts at zero b and zero c units) If the side lengths are dropped, the intercepts are 1, 0, and 0.

The reciprocals of these are also 1, 0, and 0 Since these are already the smallest integers, the Miller indices are specified by enclosing them in parentheses: (100) The commas are not included because they are simply part of the sentence structure Figure 329b shows the (110) plane that is parallel to the z axis and is a face diagonal on the top and bottom faces of the unit cell This plane intercepts the x axis at one a distance, the y axis at one b distance, and the z axis at zero c distanceThe intercepts are 1, 1, and 0, and so are the reciprocals Since these are the smallest integers, the Miller indices are specified as (110) Figure 329d shows the crystallographic plane that intercepts the x axis at 1 2 a, the y axis at one b, and the z axis at one c.

The reciprocals are therefore 2, 1, and 1, and so this plane is identified as the (211) plane Parentheses are used, as in the preceding examples, to specify a single plane or a family of parallel planes Thus (100) represents all the planes that are parallel to the yz axes and intercept the lattice structure at one a, two a, three a, etc distances Wavy brackets, or braces, are used to designate all planes in a crystal that are equivalent For example, the six face planes of a unit cell such as that in Fig 329a are (100), (010), (001), (100), (010), and (001) The notation {100} includes all these six planes..

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