Geometry

1807 Submissions

[3] viXra:1807.0463 [pdf] submitted on 2018-07-26 06:25:03

Curve Subdivision in SE(2)

Authors: Jan Hakenberg
Comments: 6 Pages.

We demonstrate that curve subdivision in the special Euclidean group SE(2) allows the design of planar curves with favorable curvature. We state the non-linear formula to position a point along a geodesic in SE(2). Curve subdivision in the Lie group consists of trigonometric functions. When projected to the plane, the refinement method reproduces circles and straight lines. The limit curves are designed by intuitive placement of control points in SE(2).
Category: Geometry

[2] viXra:1807.0298 [pdf] submitted on 2018-07-17 17:10:18

Study of Transformations

Authors: Yeray Cachón Santana
Comments: 10 Pages.

This paper covers a first approach study of the angles and modulo of vectors in spaces of Hilbert considering a riemannian metric where, instead of taking the usual scalar product on space of Hilbert, this will be extended by the tensor of the geometry g. As far as I know, there is no a study covering space of Hilbert with riemannian metric. It will be shown how to get the angle and modulo on Hilbert spaces with a tensor metric, as well as vector product, symmetry and rotations. A section of variationals shows a system of differential equations for a riemennian metric.
Category: Geometry

[1] viXra:1807.0234 [pdf] replaced on 2018-07-14 06:23:07

Making Sense of Bivector Addition

Authors: James A. Smith
Comments: 18 Pages.

As a demonstration of the coherence of Geometric Algebra's (GA's) geometric and algebraic concepts of bivectors, we add three geometric bivectors according to the procedure described by Hestenes and Macdonald, then use bivector identities to determine, from the result, two bivectors whose outer product is equal to the initial sum. In this way, we show that the procedure that GA's inventors dened for adding geometric bivectors is precisely that which is needed to give results that coincide with those obtained by calculating outer products of vectors that are expressed in terms of a 3D basis. We explain that that accomplishment is no coincidence: it is a consequence of the attributes that GA's designers assigned (or didn't) to bivectors.
Category: Geometry