Affine 2D Analogue

We now seek a flow in affine geometry with similar properties as the Euclidean shortening flow. We want to derive an affine curvature deformation flow analogue to the previous section. We first give the basic concepts of affine differential geometry in the plane and then extend it to 3D affine space.

Figure 1: Basic Affine Differential Geometry

A general affine transformation is given by $Y = X A + B$, where A is a 2x2 invertible matrix with positive determinant and B is a translation vector in $R^2$. Transformations of that type form a group called group of proper affine motions [13]. In our application, we restrict the determinant of A to be equal to one to obtain an invariant transformation. Thus, from our usual orthogonal Frenet frame {$e_1, e_2$} in Euclidean differential geometry, we want a frame {$a_1, a_2$} defined at each point of the curve C(p), where $a_1$ is always tangent to the the curve. A natural choice of frame is {$C_p$, $C_{pp}$}, as $C_p$ is always tangent to C(p). However, note that for an arbitrary parametrization (not arc-length), $C_{pp}$ is not necessarily orthogonal to $C_p$. Hence, {$a_1, a_2$} $=$ {$C_p$, $C_{pp}$} defines an oblique frame, as seen in Figure 1. Now, we look for a parametrization s, the affine arc-length, of the curve such that the area of the parallelogram defined by this oblique frame is one. We will then have an area-preserving affine transformation of the Frenet frame.

Let $[C,C']$ denote the 2x2 determinant where the first column is defined by components of C and the second column by those of C', i.e.,

$$[C,C']= \left \vert \begin{array}{cc} x & x' \\ y & y' \end{array} \right \vert$$

Hence, we define a parameter s such that the area of the parallelogram is one, $[C_s , C_{ss}] = 1$. Using the chain rule, we obtain the following derivation,

$$\begin{array}{ccl} [C_p, C_{pp}] &=& x_py_{pp} - x_{pp}y_p \... ..., C_{ss}] \\ &=& \left ( \frac{ds}{dp} \right )^3 \end{array}$$ (1)

From this derivation, we get the affine metric

$$g = [C_p, C_{pp}]^\frac{1}{3}$$ (2)

and the affine arc-length parameter

$$s(p) = \int _0^p g(\xi)d\xi$$ (3)

We also define the affine tangent to be $C_s$, the affine normal $C_{ss}$ and the affine curvature to be $\mu = [C_{ss}, C_{sss}]$ [13]. From equation (2), (3) and the chain rule we easily get the following relevant relations

$$ds = gdp$$ (4)

$$C_s = C_p \frac{dp}{ds}$$ (5)

$$\begin{array}{ccl} C_{ss} &=& C_{pp}\left ( \frac{dp}{ds} \r... ... \\ &=& C_{pp}\frac{1}{g^2} - C_p \frac{g_p}{g^3} \end{array}$$ (6)

Now, we investigate the affine analogue of the curve shortening flow. In Euclidean differential geometry, the flow is $\mathcal{C}_t = \kappa \overrightarrow N = C_{ss}$, where s is the arc-length parameter [13]. In affine geometry, this motivates the choice of evolution $\mathcal{C}_t = C_{ss}$, where s is now the affine arc-length. However, we must deal with the fact that basic affine differential geometry is only defined for convex curve as the affine metric g cannot equal zero. If so, the affine normal is undefined and all definitions collapse. For non-convex curves, g only vanishes at inflection points of a curve. Thus, we modify the flow as proposed in [8] and [9]

$$\mathcal{C}_t = \left\{ \begin{array}{ll} 0 & \textrm{at in... ...n points} \\ C_{ss} & \textrm{otherwise} \end{array} \right.$$ (7)

Now, we want to express the affine normal vector, $C_{ss}$, in terms of Euclidean quantities. We use equation (6) to do so. We choose the parameter p to be the Euclidean arc-length. Thus, $C_p = \overrightarrow T$ and $C_{pp} = \kappa \overrightarrow N$, where $\overrightarrow T$ and $\overrightarrow N$ are the Euclidean unit tangent and normal respectively. Hence,

$$\begin{array}{ccl} g &=& [C_p , C_{pp}]^{\frac{1}{3}}\\ &=... ...errightarrow N$ are orthogonal unit length vectors} \end{array}$$

Therefore, we have

$$\begin{array}{ccl} C_{ss} &=& C_{pp}\frac{1}{g^2} - C_p \fra... ...kappa _p}{3\kappa ^{\frac{5}{3}}} \overrightarrow T \end{array}$$ (8)

It is well-known in curve evolution theory that the tangential component of the velocity vector affects only the parametrization of the family of curves in the evolution, not their shape [7]. This fact combined with the fact that curvature, $\kappa$, vanishes at inflection points, implies that the affine flow given by (7) is geometrically equivalent to

$$\mathcal{C}_t = \kappa^{\frac{1}{3}} \overrightarrow N$$ (9)

This flow is what all the the literature on affine evolutions claim to be the affine analogue of the curve shortening flow in Euclidean space. What is the analogy? Siddiqi et. al. [15] have showed that curvature deformation in the normal direction is the fastest way to shrink the length functional of a curve in Euclidean geometry. Now, in affine geometry, the claimed analogue is the functional is affine length and the fastest way to shrink it is the evolution in the affine normal direction which corresponds to an Euclidean curvature deformation in the Euclidean normal direction.

We have strong issues with this claim and the way it is proved in [7]. First, we consider the derivation on p.105 of [7]. We carefully verified that every step of the proof is valid. It is indeed correct until the last statement

$$\begin{array}{ccl} L'(t) &=& \oint g dp \\ & \vdots & \\ &=& -\frac{2}{3}\int _0^{L(t)} \mu [C_s, C_t]ds \end{array}$$ (10)

Now, Sapiro and Tannenbaum state ``if $\mathcal{C}_t = C_{ss}$, we obtain [ $C_s, \mathcal{C}_t$] $=$ 1 , which gives the direction of the most rapid decrease for the affine perimeter''. The first part is true but the last part is false. From derivation (10), the gradient flow which will decrease the affine length as quickly as possible is the one that will maximize the area of the parallelogram defined by [$C_s,C_t$]. Setting $\mathcal{C}_t$ in the direction of the affine normal is not maximizing that area. What is the maximum area enclosed by two vectors, $\overrightarrow v$ and $\overrightarrow u$? It is maximum when the sine of the angle between the two vectors is one as

$$[\overrightarrow v, \overrightarrow u]= \vert u\vert\vert v\vert sin\theta_{uv}$$

Hence, $\overrightarrow v$ and $\overrightarrow u$ have to be orthogonal. However, we know that $C_{ss}$ and $C_s$ are not perpendicular. Therefore, something is wrong when they say that setting $\mathcal{C}_t = C_{ss}$ gives the direction of most rapid decrease of the affine perimeter.

Lets compute the expression in the integral of equation (10). We need the straight forward fact that

$$[C_s, C_{ss}]= 1 \Longrightarrow [C_s, C_{sss}] = 0 \Longrightarrow x_sy_{sss} = y_sx_{sss}$$ (11)

So,

$$\begin{array}{ccl} \mu [\mathcal{C}_t, C_s] &=& [C_{ss}, C_{... ...&=& \mu \qquad \textrm{if $\mathcal{C}_t = C_{ss}$} \end{array}$$ (12)

Hence, we believe that the statement should be corrected and stated as ``setting the curve evolution corresponding to the gradient flow $\mathcal{C}_t = C_{ss}$ decreases the affine length according to its affine curvature. In this sense, the affine version is analogue to the Euclidean flow shrinking a curve by its Euclidean curvature. It is not correct to claim, as in most papers [8-14], that the analogy between Euclidean and the affine geometric heat equation is based on the shortening flow. We have argued that the affine curve evolution is not necessarily the fastest way to shrink the affine perimeter.