Geometry

1905 Submissions

[7] viXra:1905.0353 [pdf] submitted on 2019-05-18 07:00:19

From Pythagorean Theorem to Cosine Theorem.

Authors: Jesús Álvarez Lobo
Comments: 2 Pages.

Easy and natural demonstration of the cosine theorem, based on the extension of the Pythagorean theorem.
Category: Geometry

[6] viXra:1905.0248 [pdf] submitted on 2019-05-16 19:31:52

Solution of a Vector-Triangle Problem Via Geometric (Clifford) Algebra

Authors: James A. Smith
Comments: 3 Pages.

As a high-school-level application of Geometric Algebra (GA), we show how to solve a simple vector-triangle problem. Our method highlights the use of outer products and inverses of bivectors.
Category: Geometry

[5] viXra:1905.0219 [pdf] submitted on 2019-05-16 04:19:51

The Flow of Chern

Authors: Antoine Balan
Comments: 1 page, written in english

We propose a flow over a Kaehler manifold, called the Chern flow.
Category: Geometry

[4] viXra:1905.0217 [pdf] submitted on 2019-05-14 07:31:01

Three, Four and N-Dimensional Swastikas & their Projections

Authors: Sascha Vongehr
Comments: 10 pages, Six Figures, Keywords: Higher Dimensional Geometry; Hyper Swastika; Reclaiming of Symbols; Didactic Arts

Difficulties with generalizing the swastika shape for N dimensional spaces are discussed. While distilling the crucial general characteristics such as whether the number of arms is 2^N or 2N, a three dimensional (3D) swastika is introduced and then a construction algorithm for any natural number N so that it reproduces the 1D, 2D, and 3D shapes. The 4D hyper swastika and surfaces in its hypercube envelope are then presented for the first time.
Category: Geometry

[3] viXra:1905.0088 [pdf] submitted on 2019-05-05 17:04:39

Via Geometric (Clifford) Algebra: Equation for Line of Intersection of Two Planes

Authors: James A. Smith
Comments: 6 Pages.

As a high-school-level example of solving a problem via Geometric Algebra (GA), we show how to derive an equation for the line of intersection between two given planes. The solution method that we use emphasizes GA's capabilities for expressing and manipulating projections and rotations of vectors.
Category: Geometry

[2] viXra:1905.0030 [pdf] submitted on 2019-05-02 22:25:09

Foundations of Conic Conformal Geometric Algebra and Simplified Versors for Rotation, Translation and Scaling

Authors: Eckhard Hitzer, Stephen J. Sangwine
Comments: 15 Pages. submitted to Topical Collection of Adv. in Appl. Clifford Algebras, for Proceedings of FTHD 2018, 21 Feb. 2019, 1 table, 1 figure.

This paper explains in algebraic detail how two-dimensional conics can be defined by the outer products of conformal geometric algebra (CGA) points in higher dimensions. These multivector expressions code all types of conics in arbitrary scale, location and orientation. Conformal geometric algebra of two-dimensional Euclidean geometry is fully embedded as an algebraic subset. With small model preserving modifications, it is possible to consistently define in conic CGA versors for rotation, translation and scaling, similar to [https://doi.org/10.1007/s00006-018-0879-2], but simpler, especially for translations. Keywords: Clifford algebra, conformal geometric algebra, conics, versors. Mathematics Subject Classification (2010). Primary 15A66; Secondary 11E88, 15A15, 15A09.
Category: Geometry

[1] viXra:1905.0026 [pdf] replaced on 2019-05-11 06:44:00

Cubic Curves and Cubic Surfaces from Contact Points in Conformal Geometric Algebra

Authors: Eckhard Hitzer, Dietmar Hildenbrand
Comments: 11 Pages. accepted for M. Gavrilova et al (eds.), Proceedings of Workshop ENGAGE 2019 at CGI 2019 with Springer LNCS, April 2019, 1 table, corrections: 03+11 May 2019.

This work explains how to extend standard conformal geometric algebra of the Euclidean plane in a novel way to describe cubic curves in the Euclidean plane from nine contact points or from the ten coefficients of their implicit equations. As algebraic framework serves the Clifford algebra Cl(9,7) over the real sixteen dimensional vector space R^{9,7}. These cubic curves can be intersected using the outer product based meet operation of geometric algebra. An analogous approach is explained for the description and operation with cubic surfaces in three Euclidean dimensions, using as framework Cl(19,16). Keywords: Clifford algebra, conformal geometric algebra, cubic curves, cubic surfaces, intersections
Category: Geometry