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The mechanical properties of polymers are determined by the types of forces acting between the molecules. The polymers are amorphous with random chain orientations while in the liquid state. This structure can be retained when the polymer is cooled rapidly to the solid state. In this condition, the polymer is quite isotropic. However, with slow cooling or plastic deformation, such as stretching or extruding, the molecules can become aligned. That is, the long axes of the chains of all the molecules tend to be parallel. A material in this condition is said to be oriented or crystalline, the degree of orientation being a measure of the crystallinity.When the molecular chains of a polymer have this type of directionality, the mechanical properties are also directional and the polymer is anisotropic. The strength of an aligned polymeric material is stronger along the axis of the chains and much lower in the perpendicular directions. This is due to the fact that only weak van der Waals forces hold the individual, aligned macromolecules together, whereas the atoms along the axes of the chains are held together by strong and covalent bonds. The intermolecular strength of linear polymers can be increased by the addition of polar (dipole) groups along the length of the chain. The most frequently used polar groups are chlorine, fluorine, hydroxyl, and carboxyl. The thermosetting (cross-linked) types of polymers have all the macromolecules connected together in three directions with strong covalent bonds. Consequently, these polymers are stronger than thermoplastic ones, and they are also more isotropic.

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Assume that the lights around your scene are not changing, so you can create a spherical environment map offline and use it every frame during your rendering. Figure 11-4 shows an example of a spherical environment map. Notice that the object itself is usually not part of the environment map, just the objects and lights surrounding it.

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32.3.4 Crystalline Solids Crystalline solids are by far the most frequently used ones on the basis of mechanical properties or load-carrying capacity. Moreover, of all the crystalline solids, metals are the most important. A crystal (or crystalline solid) is an orderly array of atoms having a repeating linear pattern in three dimensions. The atoms are represented as spheres of radius r. A space lattice is the three-dimensional network of straight lines that connects the centers of the atoms along three axes. The intersections of the lines are lattice points, and they designate the locations of the atoms. Although the atoms vibrate about their centers, they occupy the fixed positions of the lattice points. Figure 32.6 is a sketch of a space lattice, with the circles representing the centers of the atoms. A space lattice has two important characteristics: (1) The space-lattice network divides space into equal-sized prisms whose faces contact one another in such a way that no void spaces are present, and (2) every lattice point of a space lattice has identical surroundings. The individual prisms that make up a space lattice are called unit cells. Thus a unit cell is the smallest group of atoms which, when repeated in all three directions, make up the space lattice, as illustrated by the dark-lined parallelepiped in Fig. 32.6.

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FIGURE 32.6 A space lattice. (a) A unit cell is marked by the heavy lines. Black circles are on the front face; horizontal shading on the top face; vertical shading on the right side face; hidden circles are white. (b) An isolated unit cell showing dimensions a, b, and c and angles , , and .

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Ultimately, what you want is to generate a second set of texture coordinates that indexes into the right place in the environment map based on the eye position and on the surface normal. If the surface were a perfect mirror, what would you see To find out, you could cast a ray from the eye to the surface and then bounce it off the surface taking the normal into account. This new vector is called a reflection vector (see Figure 11-5).

Only 14 different space lattices and 7 different systems of axes are possible. Most of the metals belong to three of the space-lattice types: face-centered cubic, bodycentered cubic, and hexagonal close-packed. They are listed in Table 32.1, along with four metals that have a rhombohedral and two that have orthorhombic structures.

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