THE CONFORMAL NEO-EULERIAN ORIENTATION MAP

The rotation vector C = I 2 tan1/4-omega (where I is the unit vector designating an axis of rotation and omega 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 = I tan1/2-omega 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 2-pi: or, in an extended map a full circle representing a rotation of 4-pi. 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.