The conformal neo-eulerian orientation map

Abstract

The rotation vector C = I2 tan¼ω (where I is the unit vector designating an axis of rotation and ω the angle of rotation about that axis) yields a conformal orientation map, at least in the sense that transformations by change of reference orientation are conformal transformations. C is equal to the Rodrigues vector R = Itan½ω multiplied by the re-scaling factor g (R) = 2/[1+(1+R²)^1/2]. In this map the orbit for rotation about a fixed axis is a circular arc, spanning a diameter of a sphere of radius 2, representing a rotation of 4π: or, in an extended map a full circle representing a rotation of 4π. The locus of points representing orientations angularly equidistant from two given orientations is, in general, in the extended map, a pair of orthogonally intersecting spheres. A crystallographic example and a metallurgical application are given.