ABSTRACT
In this chapter, after a short review of crystal structures and symmetries (Section 2.2), the main mathematical parameters that are used to describe an orientation are de ned and explained. These are as follows:
Orientation matrix (Section 2.3)• “Ideal orientation,” that is, Miller or Miller-Bravais indices • (Section 2.4) Euler angles (Section 2.6)• Angle/axis of rotation (Section 2.7)• Rodrigues vector (Section 2.8)•
All these descriptors are employed to process and represent different aspects of macrotexture and microtexture measurements. This chapter will elucidate the theoretical aspects of these parameters and their interrelationships. For graphical representation, it is usual to plot orientations, represented as projected poles, Euler angles, angle/axis, or the Rodrigues vector, in an appropriate “space.” These spaces-the pole gure or inverse pole gure “projected space,” Euler space, cylindrical angle/axis space, and Rodrigues space-are explained in Sections 2.5 through 2.8. These explanations encompass only the basic theoretical aspects; in practice, their utilization is customized according to whether they are employed to display and analyze macrotexture or to display and analyze microtexture data. Therefore, these two divisions are discussed separately, in Chapters 5 and 9.
