In this article we describe the notion of affinely indepedent subset of a real linear space. First we prove selected theorems concerning operations on linear combinations. Then we introduce affine indepedence 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 al...

Introduction 3 1. Affine geometry 4 1.1. Affine spaces 5 1.1.1. Euclidean geometry and its isometries 5 1.1.2. Affine spaces 7 1.1.3. Affine transformations 8 1.1.4. Tangent spaces 9 1.1.5. Acceleration and geodesics 10 1.1.6. Connections 11 1.2. The hierarchy of structures 11 1.3. Affine vector fields 12 1.4. Affine subspaces 13 1.5. Volume in affine geometry 14 1.6. Centers of gravity 14 1.7....

We prove that the period map from the Teichmüller space of polarized and marked Calabi-Yau manifolds to the classifying space of polarized Hodge structures is an embedding. The proof is based on the constructions of holomorphic affine structure and global holomorphic affine flat coordinates on the Teichmüller space.

At Pairing 2010, Lauter et al’s analysis showed that Ate pairing computation in affine coordinates may be much faster than projective coordinates at high security levels. In this paper, we further investigate techniques to speed up Ate pairing computation in affine coordinates. On the one hand, we improve Ate pairing computation over elliptic curves admitting an even twist by describing an 4-ar...

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 R 2. 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 R 2 .

In our previous work, we have generalized the notion of dually flat or Hessian manifold to quasi-Hessian manifold; it admits metric be degenerate but possesses a particular symmetric cubic tensor (generalized Amari-Centsov tensor). Indeed, naturally appears as singular model in information geometry and related fields. A is locally accompanied with possibly multi-valued potential its dual, whose...

We give new examples of affine sufaces whose rings of coordinates are d-simple and use these examples to construct simple nonholonomic Dmodules over these surfaces.

The known factorization algorithm for the maximum likelihood affine reconstruction requires that all the feature points used must be visible in all views. We derive here a closed-form-expression for the 3D coordinates of the feature points and translation vectors given the inhomogeneous affine projection matrices, but no single feature point is required to be visible in all views. The expressio...

Image and geometry processing applications estimate the local geometry of objects using information localized at points. They usually consider information about the tangents as a side product of the points coordinates. This work proposes parabolic polygons as a model for discrete curves, which intrinsically combines points and tangents. This model is naturally affine invariant, which makes it p...

An affine manifold is a differentiable manifold together with an atlas of coordinate charts whose coordinate changes extend to affine automorphisms of Euclidean space. These charts are called atline coordinates. A map between affine manifolds is called affine it its expression in affine coordinates is the restriction of an aftine map between vector spaces. Thus we form the category of affine ma...