Posts filed under ‘curves’
I had to give a talk to high school students about some mathematical notion: I decided to tell them something about elliptic curves, but not the usual speech about cryptography, finite fields and the group law on a cubic curve…
Instead, I talked about the perhaps less known appearances of elliptic functions as solutions of classical ODEs (even if I don’t really know much about these myself). The simplest mechanical system whose motion is governed by an elliptic curve is the pendulum: the reason for this is that the ODE which classically describes the time evolution of the angle of the pendulum is best rewritten in terms of the altitude of the pendulum: the law of energy conservation is then written as
where 0 and 2l are the extremal values of the altitude , is the highest altitude which can be reached with a given energy (even if , which corresponds to the pendulum make full rotations around its axis), and is the vertical momentum of the pendulum.
In this setting, there are classical Hamilton relations , , so the differential form turns out to be the canonical non-vanishing abelian differential on the elliptic curve. This explains why the period of the pendulum is an elliptic integral, which can be calculed by an arithmetic-geometric mean, and why the position of the pendulum at can be deduced from its position at times and by the classical secant-tangent law.
The notes for the talk (in French) are available here.
Le calcul de la moyenne arithmético-géométrique est un algorithme simple et très puissant découvert par Gauss permettant de calculer les intégrales elliptiques. L’excellent article de David Cox  dans L’enseignement mathématique expose l’historique des découvertes et recherches de Gauss sur le sujet.
La moyenne arithmético-géométrique a été définie par Lagrange, et est calculée de la manière suivante : si a et b sont deux réels positifs, on définit leur moyenne géométrique et leur moyenne arithmétique . Il est bien connu que , et on a
Cette relation permet de montrer que si on définit des suites et telles que et sont les moyennes arithmétique et géométrique de et , ces suites convergent vers une même limite, notée , la moyenne arithmético-géométrique.
Gauss remarqua à l’aide d’un changement de variable (très) astucieux que l’intégrale
restait invariante en remplaçant a et b par leurs moyennes arithmétique et géométrique. Il en déduit la valeur . Il montra aussi que la longueur d’un quart de lemniscate de Bernoulli est .
On sait également qu’il relia ces propriétés à celles des fonctions thêta.
Working in the Euclidean (projective) plane, a bicircular quartic curve is defined to be a quartic which is singular at the circular points I and J. We are usually interested in real curves, so the type of singulaity is the same at I or J. Salmon in his Treatise on Higher Plane Curves and Basset in his Elementary Treatise on Cubic and Quartic curves deal in detail with these curves.
A plane quartic curve has arithmetic genus , but since bicircular quartics have at least two double points, they have geometric genus 0 or 1 (genus of the desingularized curve).
Curves having geometric genus 0 are called rational curves (formerly known as unicursal curves), and admit a parameterization by rational functions of one variable.
Families of bicircular curves were defined by Cassini and Descartes by metric properties : Cassini ovals are defined by the equation where A and B are fixed foci, while Descartes define Cartesian ovals by the equation (these equations are equivalently to algebraic quartic equations, using appropriate squarings).
Inversion is a transformation of the Euclidean plane closely related to circles. It is seldom taught in high schools (at least in France), but it has special interest when dealing with circles, since it maps circles and lines to circles or lines. Inversion is defined with respect to a center O and a radius ρ (in French, the inversion is said to have pôle O and puissance (power) ρ²).