Consistent computation of first- and second-order differential quantities for surface meshes

Differential quantities, including normals, curvatures, principal directions, and associated matrices, play a fundamental role in geometric processing and physics-based modeling. Computing these differential quantities consistently on surface meshes is important and challenging, and some existing methods often produce inconsistent results and require ad hoc fixes. In this paper, we show that the computation of the gradient and Hessian of a height function provides the foundation for consistently computing the differential quantities. We derive simple, explicit formulas for the transformations between the first- and second-order differential quantities (i.e., normal vector and principal curvature tensor) of a smooth surface and the first- and second-order derivatives (i.e., gradient and Hessian) of its corresponding height function. We then investigate a general, flexible numerical framework to estimate the derivatives of the height function based on local polynomial fittings formulate...

• Type: Journal
• Year: 2008
• Where: SMA
• Authors: Xiangmin Jiao, Hongyuan Zha
• Keywords: Corresponding Height Function, Height Function, Polynomial Fitting, SMA 2008, Solid Modeling