Evolution Properties of Space Curves

The Curvature Scale Space and Torsion Scale Space Images of a space curve are a multi-scale representation for that curve which satisfies several criteria for shape representation and is therefore a preferred representation method for space curves. The torsion scale space image of a space curve is computed by convolving a path-based parametric representation of the curve with Gaussian functions of varying widths, extracting the torsion zero-crossings of the convolved curves and combining them in a torsion scale space image of the curve. The curvature scale space image of the curve is computed similarly but curvature level-crossings are extracted instead. An evolved version of a space curve $\Gamma $ is obtained by convolving a parametric representation of that curve with a Gaussian function of variance $\sigma ^{2}$ and denoted by $\Gamma _{\sigma }$. The process of generating the ordered sequence of curves $ \{ \Gamma _{\sigma }\mid \sigma \geq 0\} $ is referred to as the {\it evolution} of $\Gamma $. A number of evolution properties of space curves are investigated in this paper. It is shown that the evolution of space curves is invariant under rotation, uniform scaling and translation of those curves. This property makes the representation suitable for recognition purposes. It is also shown that properties such as connectedness and closedness of a space curve are preserved during evolution of the curve and that the center of mass of a space curve remains the same as the curve evolves. Among other results is the fact that a space curve contained inside a simple, convex object, remains inside that object during evolution. The two main theorems of the paper examine a space curve during its evolution just before and just after the formation of a cusp point. It is shown that strong constraints on the shape of the curve in the neighborhood of the cusp point exist just before and just after the formation of that point.