Affine Curvature

Conics

Suppose that β(s) is a curve parameterized by special affine arclength with constant affine curvature k. Let

Note that det C_β, since β is assumed to carry the special affine arclength parameterization, and that

It follows from the form of C_β that

By applying a suitable special affine transformation, we can arrange that C_β(0) = I is the identity matrix. Since k is constant, it follows that C_β is given by the matrix exponential

$\begin{align} C_\beta(s) &= \exp\left\{s\cdot\begin{bmatrix}0&-k\\1&0\end{bmatrix}\right\}\\ &=\begin{bmatrix}\cos\sqrt{k}s&\sqrt{k}\sin\sqrt{k}s\\ -\frac{1}{\sqrt{k}}\sin\sqrt{k}s&\cos\sqrt{k}s\end{bmatrix}. \end{align}$

The three cases are now as follows.

k = 0

If the curvature vanishes identically, then upon passing to a limit,

so β'(s) = (1,s), and so integration gives

up to an overall constant translation, which is the special affine parameterization of the parabola y = x2/2.

k > 0

If the special affine curvature is positive, then it follows that

so that

up to a translation, which is the special affine parameterization of the ellipse kx2 + k2y2 = 1.

k < 0

If k is negative, then the trigonometric functions in C_β give way to hyperbolic functions:

$C_\beta(s) =\begin{bmatrix}\cosh\sqrt{|k|}s&\sqrt{|k|}\sinh\sqrt{|k|}s\\ \frac{1}{\sqrt{|k|}}\sinh\sqrt{|k|}s&\cosh\sqrt{|k|}s\end{bmatrix}.$

Thus

up to a translation, which is the special affine parameterization of the hyperbola

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Affine Curvature - Conics