New Aspects of Geometrical Calculus With Invariants (original) (raw)

Five Lectures on Projective Invariants

We introduce invariant rings for forms (homogeneous polynomials) and for d points on the projective space, from the point of view of representation theory. We discuss several examples, addressing some computational issues. We introduce the graphical algebra for the invariants of d points on the line. This is an expanded version of the notes for the School on Invariant Theory and Projective Geometry, Trento, September 17-22, 2012. Le teorie vanno e vengono ma le formule restano. 2 G.C. Rota

Invariant operators on geometric quantities

Journal of Soviet Mathematics, 1982

The operations on geometric objects presently known which commute with changes of variables are described, and their properties are discussed.

The shape invariant of triangles and trigonometry in two-point homogeneous spaces

Geometriae Dedicata, 1990

We define a fourth basic invariant, which, besides the lengths of the three sides of a triangle, determines a triangle in the complex and quaternion projective spaces ℂP n and ℍP n (n≥2) uniquely up to isometry. We give inequalities describing the exact range of the four basic invariants. We express the angular invariants of a triangle with our basic invariants, giving a new completely elementary proof of the laws of trigonometry. As a corollary we derive a large number of congruence theorems. Finally we get, in exactly the same way, the corresponding results for triangles in the complex and quaternion hyperbolic spaces ℂH n and ℍH n (n≥2).

2. Generalized Homogeneous Coordinates for Computational Geometryy

2001

The standard algebraic model for Euclidean space E n is an n-dimensional real vector space R n or, equivalently, a set of real coordinates. One trouble with this model is that, algebraically, the origin is a distinguished element, whereas all the points of E n are identical. This deficiency in the vector space model was corrected early in the 19th century by removing the origin from the plane and placing it one dimension higher. Formally, that was done by introducing homogeneous coordinates [H91]. The vector space model also lacks adequate representation for Euclidean points or lines at infinity. We solve both problems here with a new model for E n employing the tools of geometric algebra. We call it the homogeneous model of E n . Our "new model" has its origins in the work of F. A. Wachter (1792-1817), a student of Gauss. He showed that a certain type of surface in hyperbolic geometry known as a horosphere is metrically equivalent to Euclidean space, so it constitutes a non-Euclidean model of Euclidean geometry. Without knowledge of this model, the technique of comformal and projective splits needed to incorporate it into geometric algebra were developed by Hestenes in [H91]. The conformal split was developed to linearize the conformal group and simplify the connection to its spin representation. The projective split was developed to incorporate all the advantages of homogeneous coordinates in a "coordinate-free" representation of geometrical points by vectors.

Computation in Projective Space

MAMECTIS conference, La Laguna, Spain, WSEAS, pp.152-157, ISBN978-960-474-094-9, 2009

This paper presents solutions of some selected problems that can be easily solved by the projective space representation. If the principle of duality is used, quite surprising solutions can be found and new useful theorems can be generated as well. There are many algorithms based on computation of intersection of lines, planes, barycentric coordinates etc. Those algorithms are based on representation in the Euclidean space. Sometimes, very complex mathematical notations are used to express simple mathematical solutions. It will be shown that it is not necessary to solve linear system of equations to find the intersection of two lines in the case of E2 or the intersection of three planes in the case of E3. Plücker coordinates and principle of duality are used to derive an equation of a parametric line in E3 as an intersection of two planes. This new formulation avoids division operations and increases the robustness of computation.

Projective Geometric Computing

Applications to Projective Geometry This paper applies geometric algebra to the geometry of conics in the plane. Starting from the classical double algebra expression for a conic on 5 points in terms of a running variable, we show how to eliminate this variable (by the use of tensor products) and express the conic on 5 points without resorting to a running variable. Writing , and designating the conic by Q P , the homogene-

Computing Protective and Permutation Invariants of Points and Lines

1997

Until very recently it was believed that visual tasks require camera calibration. More recently it has been shown that various visual or visually-guided robotics tasks may b e carried out using only a projective representation characterized by the projective i n variants. This paper studies di erent algebraic and geometric methods of computation of projective invariants of points and/or lines using only informations obtained by a pair of uncalibrated cameras. We develop combinations of those projective invariants which are insensitive to permutations of the geometric primitives of each of the basic con gurations and test our methods on real data in the case of the six points con guration.

Polynomials in finite geometry

We will not be very strict and consistent in the notation (but at least we'll try to be). However, here we give a short description of the typical notation we are going to use. If not specified differently, q = p h is a prime power, p is a prime, and we work in the Desarguesian projective (or affine) space PG(n, q) (AG(n, q), resp.), each space coordinatized by the finite (Galois) field GF(q). The n-dimensional vectorspace over GF(q) will be denoted by V(n, q) or simply by GF(q) n. When discussing PG(n, q) and the related V(n + 1, q) together then for a subspace dimension will be meant rank=dim+1 projectively while vector space dimension will be called rank. A field, which is not necessarily finite will be denoted by F. In general capital letters X, Y, Z, T, ... will denote independent variables, while x, y, z, t, ... will be elements of GF(q). A pair or triple of variables or elements in any pair of brackets can be meant homogeneously, hopefully it will be always clear from the context and the actual setting. We write X or V = (X, Y, Z, ..., T) meaning as many variables as needed; V q = (X q , Y q , Z q , ...). As over a finite field of order q for each x ∈ GF(q) x q = x holds, two different polynomials, f and g, in one or more variables, can have coinciding values "everywhere" over GF(q). In this case we ought to write f ≡ g, as for univariate polynomials f (X), g(X) it means that f ≡ g (mod X q − X) in the ring GF(q)[X]. However, as in the literature f ≡ g is used in the sense 'f and g are equal as polynomials', we will use it in the same sense; though simply f = g and f (X) = g(X) may denote the same. Throughout this book we mostly use the usual representation of PG(n, q). This means that the points have homogeneous coordinates (x, y, z, ..., t) where x, y, z, ..., t are elements of GF(q). The hyperplane [a, b, c, ..., d] of the space have equation aX + bY + cZ + ... + dT = 0. When PG(n, q) is considered as AG(n, q) plus the hyperplane at infinity, then we will use the notation H ∞ for that ('ideal') hyperplane. If n = 2 then H ∞ is called the line at infinity ∞. According to the standard terminology, a line meeting a pointset in one point will be called a tangent and a line intersecting it in r points is an r-secant (or a line of length r). This book is about combinatorially defined (point)sets of (mainly projective or affine) finite geometries. They are defined by their intersection numbers with lines (or other subspaces) typically. The most important definitions and basic information are collected in the Glossary of concepts at the end of this book. These are: blocking sets, arcs, nuclei, spreads, sets of even type, etc. Warning. In this book a curve is allowed to have multiple components, so in fact the curves considered here are called cycles in a different terminology.