On the velocity of an implicit surface

Abstract

In this article we derive an equation for the velocity of an arbitrary time-evolving implicit surface. Strictly speaking, only the normal component of the velocity is unambiguously defined. This is because an implicit surface does not have a unique parametrization. However, by enforcing a constraint on the evolution of the normal field we obtain a unique tangential component. We apply our formulas to surface tracking and to the problem of computing velocity vectors of a motion blurred blobby surface. Other possible applications are mentioned at the end of the article.

Robust iso-surface tracking for interactive character skinning

“…They focus on the accurate approximation of a given iso-surface rather than trying to deform an existing mesh while guaranteeing the time coherence of the deformation. Tracking can also be performed by assuming a translational transformation to extract a tangential component of the unknown deformation velocity field [Stam and Schmidt 2011] using differential information. This approach does not prevent the use of a relaxation scheme, and it appears less attractive for skinning, which mostly involves rotational transformations.…”

Calculation of Velocity on an Implicit Surface by Curvature Invariance

“…In most cases, the normal velocity or the velocity given by the simulation is used to advect the mesh; however, the deformation of the mesh will be small and small features of the surface will be preserved only if we use accurate velocities. Stam and Schmidt [12] calculated the total surface velocity, including the tangential component, under the assumption that the normal direction of the implicit function does not change during a frame. This method gives the tangential velocity for the mesh advection, but it cannot deal with the rotation due to the assumption of a rigid body translation.…”

“…Their method could capture the movement of the surface mesh, but it needed a long computational time. Our method is based on the total velocity [12]. We extend the method to rotating objects by using the curvature invariance instead of the normal vector to estimate the tangential velocity.…”

“…This means that texture or color cannot be advected along with the surface, and it also causes large deformation of the mesh. Stam and Schmidt [12] proposed a mesh advection method that includes the tangential velocity. They assumed that the gradient of an implicit function would be maintained throughout the computation.…”

Velocity calculation of 2D geometric objects by use of surface interpolation in 3D

“…This was presented previously as a method of morphing (Turk and O'Brien, 1999). However, it has been described that the velocity of a time-dependent implicit surface cannot be defined unambiguously (Stam and Schmidt, 2011). In this paper, we propose a method to calculate the velocity of an implicit surface by splitting it into global translational components, global rotational components and local deformational components.…”