Splossoms: blossoming polynomials on the sphere via geometric algebra.

Abstract

We introduce splossoms, the spherical analogue of polynomial blossoms, extending Ramshaw's theory of blossoming into spherical geometry. We note that the classical blossom axioms require spherical reinterpretation: strict symmetry does not hold on S2, and we identify the appropriate spherical analogues that the splossom satisfies. Splossoms provide a structured approach to designing spherical curves, generalizing Shoemake's SLERP interpolation. In addition, we define spolynomials, iterated spherical interpolants that mirror polynomial structures such as Bézier and B-spline forms, and we study their continuity. Potential applications include robotics, CNC milling, virtual/augmented reality and computer animation, where orientation curves on spheres are central. The splossom is naturally expressed in the framework of geometric algebra (GA), where rotors and spinors generalize quaternionic rotation and make spherical curve construction both simpler and more powerful. We argue that while splines on Lie groups and algebras are of general mathematical interest, GA provides a direct and computationally efficient setting for implementation. This article is part of the theme issue 'Modern applications of geometric algebra'.