Variational Skinning of an Ordered Set of Discrete 2D Balls

Abstract

This is the accepted version of the paper.This version of the publication may differ from the final published version. Abstract. This paper considers the problem of computing an interpolating skin of a ordered set of discrete 2D balls. By construction, the skin is constrained to be C 1 continuous, and for each ball, it touches the ball at a point and is tangent to the ball at the point of contact. Using an energy formulation, we derive differential equations that are designed to minimize the skin's arc length, curvature, or convex combination of both. Given an initial skin, we update the skin's parametric representation using the differential equations until convergence occurs. We demonstrate the method's usefulness in generating interpolating skins of balls of different sizes and in various configurations. Permanent

Skinning of circles and spheres

“…A recent approach to the skinning problem for circles and spheres is Slabaugh's method (Slabaugh et al,2008;Slabaugh et al,2009). It is an iterative way to construct the desired curves or surfaces.…”

“…The final positions of these points and tangents are obtained by the end of several iteration steps. (Slabaugh et al,2008)). For two-sided skin they constrain the points of contact to be separated by 180 degrees, but this way some of the points of contact may fall into other circles (middle figure).…”

“…Moreover, the process has to be restarted after any modification of data, thus this method is not suitable for real time modeling and adjustment. For comparison to our method: the one-sided skin in Figure 4 (left) has been computed by Slabaugh in 143 milliseconds ( (Slabaugh et al,2008), the two-sided skin (right) has been computed by our method in 14 milliseconds (both at single core 3GHz CPU).…”

“…After precisely defining the admissible configuration of initial circles and the desired geometric properties of the skin, we construct the future touching points and tangents of the skin by applying classical geometric methods, like cyclography and the ancient problem of Apollonius, finding touching circles of three given circles. Comparing the proposed method to a recent technique (Slabaugh et al,2008;Slabaugh et al,2009), larger class of admissible data set and fast computation are the main advantages. Spatial extension of the problem for skinning of spheres by a surface is also discussed in detail.…”

3D ball skinning using PDEs for generation of smooth tubular surfaces

“…This surface can then be used for visualization of the blood vessel, simulation of blood flows using computational fluid dynamics, as well as measurements such as volume or surface area. We note that the problem of 2D ball skinning was addressed in our previous work [8]; in this paper, we extend the methodology to skinning 3D balls. In this case, the problem has a similar conceptual formulation based on differential equations; however, the geometry is notably different: instead of minimizing the arc length and curvature of a curve, we minimize the surface area and mean curvature of a surface.…”