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Bernstein polynomials and B\'ezier curves play an important role incomputer-aided geometric design and numerical analysis, and their study relatesto mathematical fields such as abstract algebra, algebraic geometry andprobability theory. We describe a theoretical framework that incorporates thedifferent aspects of the Bernstein-B\'ezier theory, based on concepts fromtheoretical physics. We relate B\'ezier curves to the theory of angularmomentum in both classical and quantum mechanics, and describe physicalanalogues of various properties of B\'ezier curves -- such as their connectionwith polar forms -- in the context of quantum spin systems. This previouslyunexplored relationship between geometric design and theoretical physics isestablished using the mathematical theory of Hamiltonian mechanics andgeometric quantization. An alternative description of spin systems in terms ofharmonic oscillators serves as a physical analogue of P\'olya's urn models forB\'ezier curves. We relate harmonic oscillators to Poisson curves and theanalytical blossom as well. We present an overview of the relevant mathematicaland physical concepts, and discuss opportunities for further research.

A Physical Perspective on Control Points and Polar Forms: B\'ezier Curves, Angular Momentum and Harmonic Oscillators