## Posts tagged ‘geometry’

### Elliptic curves for high school students

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 $\ddot{x} + \sin x = 0$ 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
$p^2 = q(q-q_0)(q-2l) = P(q)$
where 0 and 2l are the extremal values of the altitude $q$, $q_0$ is the highest altitude which can be reached with a given energy (even if $q_0 > 2l$, which corresponds to the pendulum make full rotations around its axis), and $p$ is the vertical momentum of the pendulum.

In this setting, there are classical Hamilton relations $dq = p dt$, $dp = P'(q) dt$, so the differential form $dt = dq/p$ 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 $t = t_1 + t_2$ can be deduced from its position at times $t_1$ and $t_2$ by the classical secant-tangent law.

The notes for the talk (in French) are available here.

### Lines in space and the Grassmann variety

This is an English version of my previous post. Schubert calculus is a collection of techniques and formulae used for computations of enumerative, numerical properties of common objects in linear algebra (there are very nice books by William Fulton covering the subject). The example of the set of lines in three-dimensional space is commonly chosen to illustrate the kind of results which are obtained by Schubert calculus: the reason is probably that it is the simplest non trivial situation.

A line in affine space would usually by defined by its direction (which is a line in a vector space, and depends on two parameters: imagine parameterising it by a vector on a sphere), and its position, depending on two more parameters, for a given direction (think of the possible translations of a given line). Thus the set of lines might be parameterised by four parameters, and more precisely one can show that it has dimension four. Even more interestingly, two parallel planes define for almost every line a unique pair of intersection points (one on each plane) which enables us to recover the line. This means lines are (almost all) parameterised by four coordinates, by means of rational functions (well, we still have to explain what is parameterised: for example, we could say that the slope, direction of the line and its position are rational functions of the coordinates on the two planes). The resulting parameterisation is somewhat bijective, but a few lines are missing (for example, lines parallel to the plane): however, we say the set of lines is a rational variety: which means the parameterisation is essentially bijective, and the missing elements can be obtained by taking limits.

### The Plücker embedding

To avoid tiring case disjunctions, it is very nice to extend the theory to the set of lines in projective space. They can be defined by their Plücker coordinates : given a line in space, we choose two points, given by projective (a.k.a. barycentric) coordinates ${M = [a_1:a_2:a_3:a_4]}$ and ${N = [b_1:b_2:b_3:b_4]}$. Then we define the Plücker coordinates of the line $MN$ by the six numbers $x_{ij} = (a_i b_j - a_j b_i)$. The fundamental properties of these coordinates is their invariance under changing the choice of points. Any other pair of points would be given by barycentres of M and N (with weights $(\lambda,\mu)$ and $(\nu,\pi)$), giving new coordinates
$(\lambda a_i + \mu b_i)(\nu a_j + \pi b_j) - (\lambda a_j + \mu b_j)(\nu a_i + \pi b_i)$
$= (\lambda \pi - \mu \nu)(a_i b_j - a_j b_i)$
which are actually proportional to the old coordinates. These coordinates, as projective coordinates, are thus uniquely defined (and even allow to recover the line). The set of lines now appears as a subvariety of a projective space called the Grassmann variety of lines in 3-space.

### Les droites de l’espace et la grassmannienne

Le calcul de Schubert désigne un ensemble de techniques destinées à calculer les propriétés énumératives, ou numériques, d’objets communs de l’algèbre linéaire (voir notamment les excellents ouvrages de William Fulton à ce propos). L’exemple traditionnellement choisi, et probablement le plus simple, concerne l’ensemble des droites de l’espace (à 3 dimensions).

Une droite de l’espace est habituellement repérée par sa direction (qui est une droite vectorielle, et dépend donc de deux paramètres, et sa position, qui dépend de deux paramètres supplémentaires (à direction fixée). L’ensemble des droites peut donc être paramétré par quatre paramètres, on peut montrer qu’il est de dimension quatre. Plus intéressant encore, si on place dans l’espace deux plans parallèles, presque toutes les droites peuvent être décrites (de manière unique) par leurs points d’intersection avec ces plans : on obtient ainsi une description par 4 fractions rationnelles (à supposer qu’on sache ce qu’on est en train de paramétrer, ce qui sera plus clair dans une seconde). Ce paramétrage est à peu de chose près bijectif (il manque les droites un peu particulières) : on dit que l’ensemble des droites forme une variété rationnelle.

### Le plongement de Plücker

Pour éviter de fastidieuses études de cas, on s’intéresse également aux droites de l’espace projectif : on peut les repérer par les coordonnées de Plücker. Étant donnée une droite de l’espace, considérons deux points sur cette droite de coordonnées projectives ${M = [a_1:a_2:a_3:a_4]}$ et ${N = [b_1:b_2:b_3:b_4]}$. Les coordonnées de Plücker de la droite sont, par définition, les 6 nombres $x_{ij} = (a_i b_j - a_j b_i)$. Si on avait choisi d’autres points (qui seraient donc des barycentres de M et N), on aurait obtenu des nombres de la forme

$(\lambda a_i + \mu b_i)(\nu a_j + \pi b_j) - (\lambda a_j + \mu b_j)(\nu a_i + \pi b_i)$
$= (\lambda \pi - \mu \nu)(a_i b_j - a_j b_i)$

qui sont en fait proportionnels à ceux calculés avec M et N. Les droites sont donc naturellement paramétrées par des coordonnées projectives et forment la variété grassmannienne des droites de l’espace.

### Bicircular quartic curves

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 $h^1(\mathcal O_X) = 3$, 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 $MA \cdot MB = k^2$ where A and B are fixed foci, while Descartes define Cartesian ovals by the equation $a \cdot MA + b \cdot MB = k$ (these equations are equivalently to algebraic quartic equations, using appropriate squarings).

### Inversion and n-circular curves

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).

### Circles in projective geometry

Euclidean geometry is closely related to the ability to define circles, which exists naturally in two settings: the complex projective line and the real projective plane.

### The Riemann sphere

The complex projective line $\mathbb P^1(\mathbb C)$ is a natural compactification of the Euclidean plane: its natural automorphisms are homographies or Möbius transformations, $z \mapsto \frac{az+b}{cz+d}$ which transform lines or circles into lines or circles. The Euclidean geometry arises from the natural isomorphism between the group of homographies stabilizing the point at infinity and the group of direct isometries of the plane $SO_2(\mathbb R^2) \ltimes \mathbb R^2$.

### Casey’s Treatise on analytical geometry

John Casey’s Treatise on the analytical geometry of the point, line, circle and conic sections is a book from the end of the 19th century gathering a wide range of results and constructions in plane geometry known at those times. The Bibliothèque Nationale de France gives free access to a digitized version of the book (see there).

The book begins with the definitions of classical coordinate systems: cartesian coordinates, polar coordinates, projective homogeneous coordinates (which were then dubbed trilinear or barycentric). Casey has no taboo concerning points of the field of complex numbers: in particular, many statements make use of cyclic points or zero-radius circles. Much further into the book, the author introduces projections and perspectives, as well as quadratic cones, and harmonic divisions.