Centro-affine curvesBy: Gil Bor, CIMAT, gil@cimat.mx(Under construction. Last modified: .) Definitions [1]. A smoothly immersed curve $\gamma$ in $\mathbb{R}^2$ is non-radial if the tangent line at each point does not pass through the origin. In particular, $\gamma$ itself does not pass through the origin. Such curves are oriented in the anti-clockwise direction and can be parametrized, uniquely up to parameter shift, so that $[\gamma(t), \gamma'(t)]=1.$ That is, the radius vector $0\gamma(t)$ sweeps area at a constant rate $1/2$. This is called a centro-affine parametrization and is invariant under the usual linear action of $\mathrm{SL}_2(\mathbb{R})$ on $\mathbb{R}^2$. If $t$ is such a paraemeter then $\gamma''=p(t)\gamma$ for some function $p(t)$ along the curve, called its centro-affine curvature . For example, $(t, -1)$, $(\cos t, \sin t)$ and $(e^{-t}, e^{t})/\sqrt{2}$ are centro-affine paramerizations of a line $y=-1$, the circle $x^2+y^2=1$ and the hyperbola branch $xy=2$, $x,y>0$, with constant curvatures $0,-1,1$ (resp.). They are orbits of 1-parameter subgroups of $\mathrm{SL}_2(\mathbb{R})$. $c$-transform. A $c$-transform, or centro-affine Bäcklund transform, of a non-radial curve $\gamma$ is another non-radial curve $\delta$, with centro affine parametrizations $\gamma(t), \delta(t),$ such that $[\gamma(t),\delta(t)]=c$ for all $t$. Equivalently, one can slide points $A,B$ along $\gamma, \delta$ (resp.), such that the triangle $OAB$ has constant oriented area $c/2$ and the midpoint $(A+B)/2$ always moves in the direction of the segment $AB$. A non-radial curve $\gamma$ is self-Bäcklund if $A,B$ can be taken to lie on $\gamma$. Equivalently, it is its own $c$-transform, up to a centro-affine parameter shift. If $\gamma$ is closed, with centro-affine perimeter $L$, the ratio $c/L$ is called the rotation number. A self-Bäcklund curve may admit many rotation numbers. For example, the circle is self-Bäcklund for all rotation numbers in $(0,1)$ (probably the only such curve). A carousel is a closed self-Bäcklund curve with a rational rotation number $m/n$. Equivalently, an inscribed equilateral centro-affine $n$-gon, where $c/L=m/n$, can be rotated in it so that the midpoint of every edge moves in the direction of the edge. Self-Bäcklund centro-affine curves from Lamé equation. The equation $\gamma''=p(t)\gamma$, where $p(t)=A+B\wp(t)$ and $\wp$ is the Weierstrass function, is called the Lamé equation and can be solved explicitely using elliptic functions. Choosing carefully $A,B$ and the periods of $\wp$, the solutions give many examples of self-Bäcklund centroaffine curves. Namely, for each pair of relatively prime positive integers $(n,k)$, where $0 < n < k$ and $n$ is odd, there is a closed curve $\gamma_{n,k}(t)$, $2\pi$-periodic (all periods are minimal), with $2k$-fold symmetry (ie invariant under rotation by $\pi/k$ radians) and its winding number is $n$ (simple iff $n=1$). Centroaffine elasticae [2]. These are closed centro affine curves, ie $[\gamma, \gamma']=1$, critical wrt $\int p^a$, where $p:=[\gamma',\gamma'']$, among curves of fixed period (note: this is the negative of the definition of $p$ above). They satisfy the ODE $\left(p^{a-1}\right)'' = \left({2\over a} -4\right) p^a +c.$ References
Gallery$c$-transformations
A centroaffine equilateral centrally symmetric decagon. Carrousels
Self-Bäcklund curves
Recutting
Centro affine elasticae
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