Abstract
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.
Highlights
When engineers are designing the shape of an object, they are often met with a requirement on how much the surface can curve
This requirement may come from the manufacturing process, e.g. if the object is made from milling, the size of the tool naturally puts a lower bound on the radius of curvature of the surface or equivalently an upper bound on the curvature of the surface
The curvature requirement is treated as a constraint in the design process, i.e., the object at hand is designed or optimised based on some other characteristic of the object, while ensuring that the curvature of the surface complies with the prescribed value
Summary
When engineers are designing the shape of an object, they are often met with a requirement on how much the surface can curve. The curvature requirement is treated as a constraint in the design process, i.e., the object at hand is designed or optimised based on some other characteristic of the object, while ensuring that the curvature of the surface complies with the prescribed value. Since this is usually an iterative process, the method for ensuring curvature validity must be precise and efficient, in order to avoid too long design processes.
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