biVector forum

Definitions of origin and infinity inconsistent in CGA literature

I’ve been studying the 2D Conformal GA and have come across multiple inconsistencies with regard to definitions on infinity n_i, origin n_o and conformal mapping of Euclidean points and lines. All of them start with extending the generative basis with two elements, e.g., \{e, \bar{e} \} such that e^2 = 1 and \bar{e}^2 = -1. Some define the origin (infinity) as their sum (n_o = e + \bar{e}), some as their difference (n_o = e - \bar{e}). Some scale the origin by half, others scale infinity by half, others scale both. But I assume these definitions play out in the Euclidean to Conformal point mapping functions and the joining of two conformal points with a third point at infinity to achieve a conformal line. There is obviously some designers choice involved here. (An example of this problem is the ganga.js cheat-sheet for 2D conformal GA \cong \mathbb{R}_{3,1,0} which shows origin, infinity formulas different from the code to create the graphic showing points, lines, and circles.) What are the ramifications of these choices? The difficulty is that each source document I’ve looked at does not elaborate all the ramifications across the various formula, so it is hard to find a complete self-consistent set. Has anyone created a pdf cheat-sheet like @cgunn3 2DPGA.pdf so as to provide a complete self-consistent tabulation of these matters?