3D Affine Evolution

In this section, we state the 3D extension to the affine curve evolution $\mathcal{C}_t = \kappa ^\frac{1}{3} \overrightarrow N$ discussed previously. We want to find an affine deformation flow based on the principal curvatures of the surface. The 3D affine differential geometry and setting up the problem is more complicated than the the 2D case. Section 4 and 5 of [10] go through the main steps of the derivation. It turns out that the simplest possible affine surface evolution is moving every point of the surface according to the Gaussian curvature raised to some exponent. The Gaussian curvature of a surface is the product of the two principal curvature, $\kappa = \kappa _1 \kappa _2$. Why the Gaussian curvature? Analogous to the inflection points issue in 2D, it resolves the ``non-existence'' problem of 3D affine differential geometry at parabolic points, where one of the two principal curvature vanishes. Thus, using the Gaussian curvature in the flow as opposed to mean curvature assures that parabolic points stay fixed. Also, as we will see in Section 4, the Gaussian curvature has a very convenient and simplified level set formulation for this flow. The evolution is defined as

$$\mathcal{S}_t = \vert\kappa\vert^\frac{1}{4} \overrightarrow N$$ (13)

where $\kappa$ is the Gaussian curvature and $\overrightarrow N$ is the Euclidean normal. We take the positive part of the Gaussian curvature because it might be negative. This extension to non-convex surfaces was established and justified in [10, [2]] by Alvarez et. al..

We have defined all the necessary flows for our implementation and experiments. Table 1 summarizes the evolution equations. We now look at the important mathematical properties of these flows.

Table 1: Geometric Heat Evolution Equations in Euclidean and Affine Geometry

	Euclidean Geometry	Affine Geometry
2D	$\mathcal{C}_t = \kappa \overrightarrow N$	$\mathcal{C}_t = \kappa^{\frac{1}{3}} \overrightarrow N$
3D	$\mathcal{S}_t = \kappa_{\textrm{mean}} \overrightarrow N$	$\mathcal{S}_t = \vert\kappa_{\textrm{gaussian}}\vert^{\frac{1}{4}} \overrightarrow N$