1 research outputs found
A Physical Perspective on Control Points and Polar Forms: B\'ezier Curves, Angular Momentum and Harmonic Oscillators
Bernstein polynomials and B\'ezier curves play an important role in
computer-aided geometric design and numerical analysis, and their study relates
to mathematical fields such as abstract algebra, algebraic geometry and
probability theory. We describe a theoretical framework that incorporates the
different aspects of the Bernstein-B\'ezier theory, based on concepts from
theoretical physics. We relate B\'ezier curves to the theory of angular
momentum in both classical and quantum mechanics, and describe physical
analogues of various properties of B\'ezier curves -- such as their connection
with polar forms -- in the context of quantum spin systems. This previously
unexplored relationship between geometric design and theoretical physics is
established using the mathematical theory of Hamiltonian mechanics and
geometric quantization. An alternative description of spin systems in terms of
harmonic oscillators serves as a physical analogue of P\'olya's urn models for
B\'ezier curves. We relate harmonic oscillators to Poisson curves and the
analytical blossom as well. We present an overview of the relevant mathematical
and physical concepts, and discuss opportunities for further research.Comment: 22 pages, 14 figures. Comments are welcome