[6] **viXra:1909.0643 [pdf]**
*submitted on 2019-09-30 08:27:13*

**Authors:** Antoine Balan

**Comments:** 2 pages, written in english

We define a generalization of the De Rham cohomology depending of a smooth function.

**Category:** Geometry

[5] **viXra:1909.0638 [pdf]**
*submitted on 2019-09-30 11:35:00*

**Authors:** Yuly Shipilevsky

**Comments:** 2 Pages.

We give a geometrical proof of the formula:
arccos(cosα1cosβ3) + arccos(cosα2cosβ1) + arccos(cosα3cosβ2) = π,
α1 + β1 = π/2,
α2 + β2 = π/2,
α3 + β3 = π/2.

**Category:** Geometry

[4] **viXra:1909.0526 [pdf]**
*submitted on 2019-09-24 09:12:22*

**Authors:** Dante Servi

**Comments:** 2 Pages.

This submission is replaced by full version 1910.0086v3 found here
http://vixra.org/abs/1910.0086?ref=10844852

**Category:** Geometry

[3] **viXra:1909.0438 [pdf]**
*replaced on 2019-09-22 10:12:59*

**Authors:** Jesus Sanchez

**Comments:** 32 Pages.

The Geometric Algebra is a tool that can be used in different disciplines in Mathematics and Physics. In this paper, it will be used to show how the information of the Non-Euclidean metric in a curved space, can be included in the basis vectors of that space. Not needing any external (out of the metric) coordinate system and not needing to normalize or to make orthogonal the basis, to be able to operate in a simple manner. The different types of derivatives of these basis vectors will be shown. In a future revision, the Schwarzschild metric will be calculated just taking the derivatives of the basis vectors to obtain the geodesics in that space.
As Annex, future developments regarding GA are commented: rigid body dynamics, Electromagnetic field, hidden variables in quantum mechanics, specificities of time basis vector, 4π geometry (spin 1/2) and generalization of the Fourier Transform.

**Category:** Geometry

[2] **viXra:1909.0174 [pdf]**
*submitted on 2019-09-08 14:35:08*

**Authors:** Jan Hakenberg

**Comments:** 9 Pages.

We generalize the Ramer-Douglas-Peucker algorithm to operate on a sequence of elements from a Lie group. As the
original, the new algorithm bounds the approximation error, and has an expected runtime complexity of O(n log n).
We apply the curve decimation to data recorded from a car-like robot in SE(2), as well as from a drone in SE(3). The results
show that many samples of the original sequence can be dropped while maintaining a high-quality approximation to the
original trajectory.

**Category:** Geometry

[1] **viXra:1909.0028 [pdf]**
*submitted on 2019-09-01 08:30:19*

**Authors:** Yuly Shipilevsky

**Comments:** 1 Page.

We prove that in Cartesian Coordinate System,
the cosine of the angle between lines,
belonging to two coordinate planes and both lines go through the origin with the corresponding angles to the same common axis is equal to multiplication of cosines of these angles.

**Category:** Geometry