Posts filed under ‘euclidean geometry’

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 $SO_2(\mathbb R^2) \ltimes \mathbb R^2$.

Un peu de géométrie plane…

Commençons par un peu de géométrie élémentaire… Soit $ABC$ un triangle non dégénéré dans le plan euclidien. L’usage de coordonnées barycentriques permet de repérer un point $M$ du plan par trois nombres réels ${[x:y:z]}$ de sorte que $M$ soit le barycentre des sommets du triangle avec les poids indiqués.

Let’s begin with elementary geometry… Let $ABC$ be a non-degenerate triangle in the Euclidean plane. By using barycentric coordinates, any point $M$ in the plan can be assigned a triple of real numbers ${[x:y:z]}$, such that $M$ is the barycentre of the vertices with these weights.