2 Kac coordinates 5 2.1 Based automorphisms and affine root systems . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Torsion points, Kac coordinates and the normalization algorithm . . . . . . . . . . . . 9 2.3 μm-actions on Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.4 Principal μm-actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...

In this paper we consider planar sections and visual contours of co-dimension one affine immersions. The main theorem says that the third order Taylor expansion of the difference between the visual contour and planar section functions is exactly the cubic form. We also consider parameterizations on two dimensional affine immersions whose coordinate lines are geodesics, in one direction, and bot...

Periods of parallel exterior forms define natural coordinates on the deformation space of complete affine structures on the two-torus. These coordinates define a differentiable structure on this deformation space, under which it is diffeomorphic to R2. The action of the mapping class group of T 2 is equivalent in these coordinates with the standard linear action of SL(2, Z) on R2.

An affine manifold is a manifold with a distinguished system of affine coordinates, namely, an open covering by charts which map homeomorphically onto open sets in an affine space E such that on overlapping charts the homeomorphisms differ by an affine automorphism of E. Some, but certainly not all, affine manifolds arise as quotients Ω/Γ of a domain in E by a discrete group Γ of affine transfo...

In this article we describe the notion of affinely independent subset of a real linear space. First we prove selected theorems concerning operations on linear combinations. Then we introduce affine independence and prove the equivalence of various definitions of this notion. We also introduce the notion of the affine hull, i.e. a subset generated by a set of vectors which is an intersection of ...

We describe a linear algorithm to recover 3D affine shape/motion from line correspondences over three views with uncalibrated affine cameras. The key idea is the introduction of a one-dimensional projective camera. This converts the 3D affine reconstruction of “lines” into 2D projective reconstruction of “points”. Using the full tensorial representation of three uncalibrated 1D views, we prove ...