Quadratic Spinor Polynomials with Infinitely Many Factorizations

Abstract

Spinor polynomials are polynomials with coefficients in the even sub-algebra of conformal geometric algebra whose norm polynomial is real. They describe rational conformal motions. Factorizations of spinor polynomial corresponds to the decomposition of the rational motion into elementary motions. Generic spinor polynomials allow for a finite number of factorizations. We present two examples of quadratic spinor polynomials that admit infinitely many factorizations. One of them, the circular translation, is well-known. The other one has only been introduced recently but in a different context. We not only compute all factorizations of these conformal motions but also interpret them geometrically.

Figures (4)

• Figure 1: Hopf fibration via Villarceau circles on a hyperbolic family of torus surfaces.
• Figure 2: Elementary conformal motions: conformal rotation, translation (transversion), and scaling
• Figure 3: Circular translation and parallelogram linkage
• Figure 4: Different Dupin cyclides with the same Villarceau circle

Theorems & Definitions (1)

• Remark 1