Global Fulfilment of Curvature Constraints for Surfaces

Constraints on the surface curvature are seen in many design problems within engineering. Often, such curvature constraints are evaluated only in a finite set of points on the surface. This may lead to invalid designs as the constraints can be violated in other points. We propose a new method to check globally that the largest absolute value of the principal curvatures of a spline surface is below a prescribed value. The method exploits that the curvature validity condition can be reformulated as three polynomial expressions involving the derivatives of the surface parametrisation. These polynomials can be expressed explicitly using the Bernstein basis, and the global curvature validity can then be assessed directly via the coefficients of the three expressions. We demonstrate the applicability of the method on both a simple paraboloid, a bi-linear surface, and an industry-oriented surface representing a reflector antenna on a space-borne satellite.