Differential surface kinematics for time-varying, surface-surface floating contact interactions

Algorithms to compute surface gradients are a key part of many constraint satisfaction and constrained optimization techniques. Methods to solve nonlinear problems use the surface Jacobian to help determine the next value in iterative schemes. In the past, graphics techniques that were intended for interactive rate solutions relied on numerical approximations to evaluate the Jacobian. To overcome the inaccuracies and instabilities that result from these numerical methods, we present a fast, closed form solution for obtaining the surface Jacobian. We have created a constraint framework in which to test contact, self-assembly, and surface impact problems using the analytical formulation presented here. In this work we develop the analytical solution in the context of the floating contact problem. By generalizing and reformulating a previous result in differential surface kinematics, we show that this technique also yields a parametric coordinate update for minimum distance for haptics and virtual reality applications. The utility of this formulation for surface impact dynamics and optimal surface configuration is also shown.