Keywords: Conic sections Bézier curves Eccentricity Asymptotes Axes Foci In this paper, we address the calculation of geometric characteristics of conic sections (axes, asymptotes, centres, eccentricity, foci) given in Bézier form in terms of their control polygons and weights, making use of real and complex projective and affine geometry and avoiding the use of coordinates.

Let k[V ] be the space of functions from a finite vector space into the algebraically closure of its field of scalars. This paper describes the lattice of subspaces of k[V ] which are invariant under the affine group AGL(V ). The description provides a simple method for finding the submodule generated by any set of functions given as polynomials in the standard coordinates.

n-geocentric coordinate system known as the War Office -Wolf, Veis and Molodensky -parameter linear affine) transformation models for the study area by ers. In order to ascertain the precision of the in both WGS84 and War Office systems. The results -Badekas and Veis models parameters transform WGS84 coordinates to

In this lecture, we investigate the existence of a state feedback control and a change of coordinates that transform a nonlinear control system affine in the inputs into an equivalent linear system. The treatment corresponds to selected parts from Chapter 9 in [1], Chapter 13 in [2], and Chapter 4 in [3].

We derive an explicit method of computing the composition step in Cantor’s algorithm for group operations on Jacobians of hyperelliptic curves. Our technique is inspired by the geometric description of the group law and applies to hyperelliptic curves of arbitrary genus. While Cantor’s general composition involves arithmetic in the polynomial ring Fq[x], the algorithm we propose solves a linear...

This curve has the points (1 : 0 : 0) and (0 : 1 : 0) at infinity over any field. The affine equation is XY +Y +X = 0. The origin is a point of this curve. If (x, y) ∈ F8 is a point of this curve with nonzero coordinates, then x = 1. So 0 = xy + y + x = xy + xy + x = x[(xy) + (xy) + 1]. Let t = xy. Then t + t+ 1 = 0. So the Klein quartic has 3.7 = 21 rational points over F8 with nonzero coordin...

We present an approach for building an affine representation of an unknown curved object viewed under orthographic projection from images of its occluding contour. It is based on the observation that the projection of a point on a curved, featureless surface can be computed along a special viewing direction that does not belong to the point’s tangent plane. We show that by circumnavigating the ...

Quantum theory of field (extended) objects without a priori space-time geometry has been represented. Intrinsic coordinates in the tangent fibre bundle over complex projective Hilbert state space CP (N − 1) are used instead of space-time coordinates. The fate of quantum system modeled by the generalized coherent states is rooted in this manifold. Dynamical (state-dependent) space-time arises on...

An attempt to build quantum theory of field (extended) objects without a priori space-time geometry has been represented. Space-time coordinates are replaced by the intrinsic coordinates in the tangent fibre bundle over complex projective Hilbert state space CP (N − 1). The fate of quantum system mod-eled by the generalized coherent states is rooted in this manifold. Dynamical (state-dependent)...