We introduce an integral affine structure on a conic, through an elegant but simple geometrical product of points. This appears to have been first explained in the book “Elliptic Functions and Elliptic Integrals” of Prasolov and Solovyev in 1997, and then independently by Franz Lemmermeyer. They all realized that the famous product structure on a cubic curve actually has an analog in the degree two case! This product relies only on the affine structure of our geometry, and the crucial associativity is more or less equivalent to Pascal’s famous theorem. [Thanks to Cedric Pilatte for bringing the work of Prasolov and Solovyev to our attention!]
This product allows us to define a general group structure on a conic. It also allows us a consistent understanding of a conic vector, which leads to the natural idea of a log ladder on a conic. Pleasantly in the blue, red and green geometries these geometrical constructions have a natural algebraic formulation in terms of powers of elements in the associated complex number system. This has important consequences for the equality of signed areas, signed slice areas and signed cap areas associated to such a ladder.
In this lecture geometry, algebra and arithmetic come together, to provide an important framework on which the theories of circular, hyperbolic, logarithmic and exponential functions rest.