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Any replacements are listed farther down

[396] **viXra:1911.0268 [pdf]**
*submitted on 2019-11-15 09:20:11*

**Authors:** Antoine Balan

**Comments:** 1 page, written in english

The Riemann flow is defined with help of the riemannian curvature.

**Category:** Geometry

[395] **viXra:1910.0620 [pdf]**
*submitted on 2019-10-30 10:50:27*

**Authors:** Dante Servi

**Comments:** 8 Pages.

Further reflections related to my article published on viXra.org in the geometry group at number 1910.0086 (revision v3) File name: 1910.0086v3.pdf with the following title: Poligonal spirals with manageable inclination complete version of the discussion. ---- Note: My article to which this refers is found below, at number 1910.0086.
As for my article to which this refers, it is possible that also this article of mine is updated, to be sure to download the latest revision do not click on (pdf) but on (viXra: nnnn.nnnn), it will open the page where all the revisions are, (v1), (v2), (v3), (v...). On this page click on (v...) to download the latest revision.

**Category:** Geometry

[394] **viXra:1910.0602 [pdf]**
*submitted on 2019-10-29 06:56:33*

**Authors:** Valery Timin

**Comments:** timinva@yandex.ru, 14 pages in Russian

This paper deals with the orthonormal transformation of the vectors and tensors of the 4-dimensional Galilean space. Such transformations are transformations of rotation and transition to a moving coordinate system. Formulas and matrices of these transformations are given. The transition from one coordinate system to another, moving relative to the first, did long before the theory of relativity. The natural space for "transitions from one coordinate system to another" is the Galilean space. It is the space of classical mechanics. This paper focuses on the 4-dimensional interpretation of such transformations. В данной работе рассмотрены вопросы ортонормированного преобразования векторов и тензоров 4-мерного галилеева пространства. Такими преобразованиями являются преобразования поворота и перехода в движущуюся систему координат. Даны формулы и матрицы этих преобразований.
Переход от одной системы координат к другой, движущейся относительно первой, делали задолго до появления теории относительности. Естественным пространством для "переходов от одной системы координат к другой" является галилеево пространство. Именно оно является пространством классической механики. В данной работе сделан упор на 4-мерной интерпретации таких преобразований.

**Category:** Geometry

[393] **viXra:1910.0185 [pdf]**
*submitted on 2019-10-12 09:19:32*

**Authors:** Antoine Balan

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

A moduli space is defined for any riemannian manifold.

**Category:** Geometry

[392] **viXra:1910.0141 [pdf]**
*submitted on 2019-10-09 09:09:54*

**Authors:** Antoine Balan

**Comments:** 3 pages, written in french

A moduli space is defined over a spin manifold by mean of the Dirac operator, finite dimensionality and compactness are discussed.

**Category:** Geometry

[391] **viXra:1910.0103 [pdf]**
*submitted on 2019-10-07 08:49:06*

**Authors:** Antoine Balan

**Comments:** 2 pages, written

For a family of connections in a vector fiber bundle over a riemannian manifold, a Yang-Mills flow is defined with help of the riemannian curvature of the connections.

**Category:** Geometry

[390] **viXra:1910.0086 [pdf]**
*submitted on 2019-10-06 11:37:38*

**Authors:** Dante Servi

**Comments:** 29 Pages.

Descrizione di un tipo di spirale composto da un insieme di segmenti che con riferimento ad un punto che definisco origine hanno una inclinazione gestibile e volendo costante. Descrizione di metodo grafico e di algoritmi che permettono di realizzarlo.
Nel foglio 10/10 descrivo come realizzare una spirale poligonale che ha tutti i vertici in comune con una spirale logaritmica.
Nel foglio 10 bis descrivo come calcolare (dopo aver deciso il grado di precisione con cui si vuol seguire il percorso della logaritmica) l'inclinazione da attribuire ai segmenti destinati a realizzare la spirale poligonale.
Sempre nel foglio 10 bis affermo che il mio metodo utilizzato al contrario può essere almeno provato per studiare in un modo nuovo una curva sconosciuta. Io non posso affermare di aver studiato con il mio metodo la spirale logaritmica però ne ho ricavato l'algoritmo illustrato nel foglio 10/10.
Description of a type of spiral composed of a set of segments that with a point that I define origin have a manageable inclination and wanting to be constant. Description of graphic method and algorithms that allow to realize it.
In sheet 10/10 I describe how to make a polygonal spiral that has all the vertices in common with a logarithmic spiral.
In sheet 10 bis I describe how to calculate (after deciding the degree of precision with which we want to follow the path of the logarithmic) the inclination to be attributed to the segments destined to realize the polygonal spiral.
Also in sheet 10 bis I state that my method used on the contrary can at least be tried to study an unknown curve in a new way. I cannot claim to have studied the logarithmic spiral with my method, but I have derived the algorithm illustrated in sheet 10/10.

**Category:** Geometry

[389] **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

[388] **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

[387] **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

[386] **viXra:1909.0438 [pdf]**
*submitted on 2019-09-20 13:18:03*

**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

[385] **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

[384] **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

[383] **viXra:1908.0550 [pdf]**
*submitted on 2019-08-28 05:10:36*

**Authors:** Volker Thürey

**Comments:** 3 Pages.

We present subsets of Euclidian spaces in the ordinary plane. Naturally some informations are lost. We provide examples.

**Category:** Geometry

[382] **viXra:1908.0545 [pdf]**
*submitted on 2019-08-26 10:53:25*

**Authors:** R. Welch, G. Ray

**Comments:** 11 Pages. This is a conference paper which appeared in SIGBOVIK 2018. It is published postumously as Gene Ray died in 2015.

Let r be a single 4-phase cubic day acting completely on a meridian time class. In [8], the authors address the cubically divisible nature of earth’s rotation under the additional assumption that
y′′(0, . . . ,Γ)→1∅=∫∫∫Θ∏r′(−|qF|, . . . ,‖Y‖)dM∩−0≤2∑ε=iF( ̄v).
We show that every pairwise pseudo-divine cube is partially isometric and anti-multiply intrinsic toward a fictitious same sex time transformation. This could shed important light on the conjectures of all religions and academia. In this context, the results of [8] are highly evil.

**Category:** Geometry

[381] **viXra:1908.0460 [pdf]**
*submitted on 2019-08-23 05:44:28*

**Authors:** Todor Zaharinov

**Comments:** 11 Pages.

Given three noncollinear points P, B and C, we investigate the construction of the triangle DBC with symmedian point P.

**Category:** Geometry

[380] **viXra:1908.0366 [pdf]**
*submitted on 2019-08-17 17:05:58*

**Authors:** SzÉkely Endre

**Comments:** 5 Pages. see in "Other instructions"

In this short note an important consequence of my previous paper5 is investigated.
We reinvestigate EUCLID’S 5th postulate

**Category:** Geometry

[379] **viXra:1908.0095 [pdf]**
*submitted on 2019-08-05 10:29:36*

**Authors:** Mark Adams

**Comments:** 14 Pages.

We consider the volume of the unit edge length Snub Dodecahedron.

**Category:** Geometry

[378] **viXra:1908.0074 [pdf]**
*submitted on 2019-08-04 20:39:02*

**Authors:** Atsushi Koike

**Comments:** 14 Pages.

According to Pierre Wantzel’s proof of 1837 that the trisection of 60 degree is impossible, because the cubic equation of had a absence of a rational solution. And his proof reached already a consensus as a general opinion.
I learned from Nobukazu Shimeno’s introductory book on complex numbers that Rafael Bombelli got a rational solution from the cubic equation based on the Cardinal formula. is , which can be further replaced by .
Kentaro Yano, who introduces the trisection of angles, says that the basic equation of is . In other words, the equation that Rafael Bombelli obtained a rational number solution is the same as the equation of the trisection of the angle. On the other hand, Yano raises as an example when the angle can be divided into three equal parts. Needless to say, and . Therefore, comparing Rafael Bombelli’s solution and the equation where Yano’s angle trisection is impossible and possible, the equation for angle trisection is . And I found that if then the solution is obtained at all angles. This paper proves that.

**Category:** Geometry

[377] **viXra:1908.0020 [pdf]**
*submitted on 2019-08-01 10:58:49*

**Authors:** Hiroshi Okumura

**Comments:** 2 Pages.

We generalize a problem in Wasan geometry involving
the incircle of a triangle.

**Category:** Geometry

[376] **viXra:1908.0004 [pdf]**
*submitted on 2019-08-01 03:51:30*

**Authors:** Antoine Balan

**Comments:** 1 page, written in english

Here is defined a generalization of connections with help of exterior forms.

**Category:** Geometry

[375] **viXra:1907.0581 [pdf]**
*submitted on 2019-07-29 19:14:56*

**Authors:** Hiroshi Okumura

**Comments:** 5 Pages.

We consider a problem in Wasan geometry involving a golden arbelos.

**Category:** Geometry

[374] **viXra:1907.0547 [pdf]**
*submitted on 2019-07-27 15:33:13*

**Authors:** Valery Timin

**Comments:** language: Russian, number of pages: 6, mailto:timinva@yandex.ru, Creative Commons Attribution 3.0 License

This paper deals with the orthonormal transformation of vectors of the 4-dimensional Galilean space. Such transformations are transformations of rotation and transition to a moving coordinate system. Formulas and matrices of these transformations are given.
The transition from one coordinate system to another, moving relative to the first, did long before the theory of relativity. The natural space for "transitions from one coordinate system to another" is the Galilean space. It is the space of classical mechanics. This paper focuses on the 4-dimensional interpretation of such transformations.
В данной работе рассмотрены вопросы ортонормированного преобразования векторов 4-мерного галилеева пространства. Такими преобразованиями являются преобразования поворота и перехода в движущуюся систему координат. Даны формулы и матрицы этих преобразований

**Category:** Geometry

[373] **viXra:1907.0545 [pdf]**
*submitted on 2019-07-27 15:41:02*

**Authors:** Valery Timin

**Comments:** language: Russian, number of pages: 6, mailto:timinva@yandex.ru, Creative Commons Attribution 3.0 License

This paper deals with the definition of conjugate vectors and three types of metrics: (dt, dl, ds) in 3+1 and 4–dimensional Galilean spaces. In Galilean space it is possible to introduce three types of orthonormal metrics:
1) spatial dl2 = dri*dri,
2) time dt = dt or dt2 = dt0*dt0 and
3) wave ds2 = dt2 – dl2.
В данной работе рассмотрены вопросы определения сопряженных векторов и трех видов метрик: (dt, dl, ds) в 3+1 и 4–мерном галилеевых пространствах

**Category:** Geometry

[372] **viXra:1907.0496 [pdf]**
*submitted on 2019-07-25 09:59:45*

**Authors:** Yuji Masuda

**Comments:** 1 Page.

This shows a geometry conposed with 1,-1,∞,i,0.

**Category:** Geometry

[371] **viXra:1907.0442 [pdf]**
*submitted on 2019-07-23 14:44:30*

**Authors:** Valery Timin

**Comments:** language: Russian, number of pages: 8, mailto:timinva@yandex.ru, Creative Commons Attribution 3.0 License

This paper deals with the orthonormal transformation of the coordinates of the 3-dimensional Euclidean space. These transformations are displacement and rotation transformations. Formulas and matrices of these transformations are given.
В данной работе рассмотрены вопросы ортонормированного преобразования координат 3-мерного евклидова пространства. Такими преобразованиями являются преобразования смещения и поворота. Даны формулы и матрицы этих преобразований.

**Category:** Geometry

[370] **viXra:1907.0441 [pdf]**
*submitted on 2019-07-23 14:46:52*

**Authors:** Valery Timin

**Comments:** language: Russian, number of pages: 11, mailto:timinva@yandex.ru, Creative Commons Attribution 3.0 License

This paper deals with the orthonormal transformation of the coordinates of 3+1 - and 4-dimensional Galilean space. Such transformations are transformations of displacement, rotation, and transition to a moving coordinate system. Formulas and matrices of these transformations are given.
The reasons for writing this work and the next few are two reasons.
1. The space in which classical mechanics is defined is the Galilean space, more precisely, its 3+1-dimensional interpretation.
2. Unlike the Galilean space, which has all the properties of the space in which tensors are defined, in classical mechanics not all parameters are tensors. In this regard, it is impossible to define classical mechanics in 4-dimensional form in 4-dimensional space in a simple way.
В данной работе рассмотрены вопросы ортонормированного преобразования координат 3+1- и 4-мерного галилеева пространства. Такими преобразованиями являются преобразования смещения, поворота и перехода в движущуюся систему координат. Даны формулы и матрицы этих преобразований.

**Category:** Geometry

[369] **viXra:1906.0404 [pdf]**
*submitted on 2019-06-20 19:17:33*

**Authors:** James A. Smith

**Comments:** 17 Pages.

As a high-school-level example of solving a problem via Geometric (Clifford) Algebra, we show how to calculate the distance and direction between two points on Earth, given the locations' latitudes and longitudes. We validate the results by comparing them to those obtained from online calculators. This example invites a discussion of the benefits of teaching spherical trigonometry (the usual way of solving such problems) at the high-school level versus teaching how to use Geometric Algebra for the same purpose.

**Category:** Geometry

[368] **viXra:1906.0302 [pdf]**
*submitted on 2019-06-16 19:22:18*

**Authors:** Israel Meireles Chrisostomo

**Comments:** 7 Pages.

This problem first appeared in the American Mathematical Monthly in 1965, proposed by Sir Alexander Oppenheim. As a matter of curiosity, the American Mathematical
Monthly is the most widely read mathematics journal in the world. On the other hand, Oppenheim was a brilliant mathematician, and for the excellence of his work in mathematics,
obtained the title of “ Sir ”, given by the English to English citizens who stand out in the
national and international scenario.Oppenheim is better known in the academic world for his
contribution to the field of Number Theory, known as the Oppenheim Conjecture.

**Category:** Geometry

[367] **viXra:1906.0278 [pdf]**
*submitted on 2019-06-15 22:14:28*

**Authors:** Israel Meireles Chrisostomo

**Comments:** 5 Pages.

This problem first appeared in the American Mathematical Monthly in 1965, proposed by Sir Alexander Oppenheim. As a matter of curiosity, the American Mathematical
Monthly is the most widely read mathematics journal in the world. On the other hand, Oppenheim was a brilliant mathematician, and for the excellence of his work in mathematics,
obtained the title of “ Sir ”, given by the English to English citizens who stand out in the
national and international scenario.Oppenheim is better known in the academic world for his
contribution to the field of Number Theory, known as the Oppenheim Conjecture.

**Category:** Geometry

[366] **viXra:1906.0074 [pdf]**
*submitted on 2019-06-05 12:20:10*

**Authors:** Antoine Balan

**Comments:** 1 page, written in english

We define the notion of SUSY non-commutativ geometry as a supersymmetric theory of quantum spaces.

**Category:** Geometry

[365] **viXra:1906.0051 [pdf]**
*submitted on 2019-06-04 11:48:09*

**Authors:** Radhakrishnamurty Padyala

**Comments:** 4 Pages. 4

Galileo derived a result for the relation between the two mean proportionals of the parts and the whole of a given line segment. He derived it for the internal division of the line segment. We derive in this note, a corresponding result for the external division of a given line segment.

**Category:** Geometry

[364] **viXra:1905.0552 [pdf]**
*submitted on 2019-05-28 11:39:36*

**Authors:** Emanuels Grinbergs

**Comments:** 13 Pages. Translated from Latvian by Dainis Zeps

Translation of the article of Emanuels Grinbergs, ОБ ОДНОЙ ГЕОМЕТРИЧЕСКОЙ ВАРИАЦИОННОЙ ЗАДАЧЕ that is published in LVU Zinātniskie darbi, 1958.g., sējums XX, izlaidums 3, 153.-164., in Russian
https://dspace.lu.lv/dspace/handle/7/46617.

**Category:** Geometry

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

**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

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

**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

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

**Authors:** Antoine Balan

**Comments:** 1 page, written in english

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

**Category:** Geometry

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

**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

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

**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

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

**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

[357] **viXra:1905.0026 [pdf]**
*submitted on 2019-05-03 05:23:22*

**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.

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

[356] **viXra:1904.0494 [pdf]**
*submitted on 2019-04-25 11:47:18*

**Authors:** Antoine Balan

**Comments:** 1 page, written in french

We define a flow for hermitian manifolds. We call it the Hermite-Ricci flow.

**Category:** Geometry

[355] **viXra:1904.0418 [pdf]**
*submitted on 2019-04-21 08:23:26*

**Authors:** Timothy W. Jones

**Comments:** 7 Pages.

Expanding the root form of a polynomial for large numbers of roots can be complicated. Such polynomials can be used to prove the irrationality of powers of pi, so a technique for arriving at expanded forms is needed. We show here how roots of polynomials can generate regular polygons whose vertices considered as roots form known expanded polynomials. The product of these polynomials can be simple enough to yield the desired expanded form.

**Category:** Geometry

[354] **viXra:1904.0398 [pdf]**
*submitted on 2019-04-20 11:06:10*

**Authors:** Yogesh H. Kulkarni, Anil D. Sahasrabudhe, Muknd S. Kale

**Comments:** 4 Pages.

Computer-aided Design (CAD) models of thin-walled parts such as sheet metal or plastics are often reduced dimensionally to their corresponding midsurfaces for quicker and fairly accurate results of Computer-aided Engineering (CAE) analysis. Generation of the midsurface is still a time-consuming and mostly, a manual task due to lack of robust and automated techniques. Midsurface failures manifest in the form of gaps, overlaps, not-lying-halfway, etc., which can take hours or even days to correct. Most of the existing techniques work on the complex ﬁnal shape of the model forcing the usage of hard-coded heuristic rules, developed on a case-by-case basis. The research presented here proposes to address these problems by leveraging feature-parameters, made available by the modern feature-based CAD applications, and by effectively leveraging them for sub-processes such as simpliﬁcation, abstraction and
decomposition.
In the proposed system, at ﬁrst, features which are not part of the gross shape are removed from the input sheet metal feature-based CAD model. Features of the gross-shape model are then transformed into their corresponding generic feature equivalents, each having a proﬁle and a guide curve. The abstracted model is then decomposed into non-overlapping cellular bodies. The cells are classiﬁed into midsurface-patch generating cells, called ‘solid cells’ and patch-connecting cells, called ‘interface cells’. In solid cells, midsurface patches are generated either by offset or by sweeping the midcurve generated from the owner-feature’s proﬁle. Interface cells join all the midsurface patches incident upon them. Output midsurface is then validated for correctness. At the end, real-life parts are used to demonstrate the efﬁcacy of the approach.

**Category:** Geometry

[353] **viXra:1904.0359 [pdf]**
*submitted on 2019-04-18 15:46:35*

**Authors:** Antoine Balan

**Comments:** 1 page, written in english

We define a natural Ricci flow for connections over a Riemannian manifold.

**Category:** Geometry

[352] **viXra:1904.0328 [pdf]**
*submitted on 2019-04-16 10:22:17*

**Authors:** Ulrich E. Bruchholz

**Comments:** 5 Pages.

It is explained why the geometry of space-time, first found by
Rainich, is generally valid. The equations of this geometry,
the known Einstein-Maxwell equations, are discussed, and results
are listed. We shall see how these tensor equations can be solved.
As well, neutrosophics is more supported than dialectics. We shall
find even more categories than described in neutrosophics.

**Category:** Geometry

[351] **viXra:1904.0123 [pdf]**
*submitted on 2019-04-06 21:12:45*

**Authors:** Hiroshi Okumura

**Comments:** 3 Pages.

We generalize a problem in Wasan geometry involving an arbelos, and construct a self-similar circle pattern.

**Category:** Geometry

[350] **viXra:1904.0023 [pdf]**
*submitted on 2019-04-01 07:30:13*

**Authors:** Edgar Valdebenito

**Comments:** 3 Pages.

En esta nota mostramos dos cuestiones elementales de geometría.

**Category:** Geometry

[349] **viXra:1904.0019 [pdf]**
*submitted on 2019-04-01 09:08:00*

**Authors:** Antoine Balan

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

We define a 3-form over a spinorial manifold by mean of the curvature tensor and the Clifford multiplication.

**Category:** Geometry

[348] **viXra:1903.0566 [pdf]**
*submitted on 2019-03-31 15:59:03*

**Authors:** Saburou Saitoh

**Comments:** 19 Pages. In this paper, we will introduce the division by zero calculus in triangles and trigonometric functions as the first stage in order to see the elementary properties.

In this paper, we will introduce the division by zero calculus in triangles and trigonometric functions as the first stage in order to see the elementary properties.

**Category:** Geometry

[347] **viXra:1903.0433 [pdf]**
*submitted on 2019-03-24 19:08:38*

**Authors:** Colin James III

**Comments:** 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

The following conjecture is refuted: "[An] n-dimensional Euclidean geometry can be embedded into (n+1)-dimensional hyperbolic non Euclidean geometry. Therefore hyperbolic non Euclidean geometry and Euclidean geometry are equally consistent, that is, either both are consistent or both are inconsistent." Hence, the conjecture is a non tautologous fragment of the universal logic VŁ4.

**Category:** Geometry

[346] **viXra:1903.0317 [pdf]**
*submitted on 2019-03-17 21:21:17*

**Authors:** Colin James III

**Comments:** 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We prove two parallel lines are tautologous in Euclidean geometry. We next prove that non Euclidean geometry of Lobachevskii is not tautologous and hence not consistent. What follows is that Riemann geometry is the same, and non Euclidean geometry is a segment of Euclidean geometry, not the other way around. Therefore non Euclidean geometries are a non tautologous fragment of the universal logic VŁ4.

**Category:** Geometry

[345] **viXra:1903.0244 [pdf]**
*submitted on 2019-03-12 10:13:23*

**Authors:** Antoine Balan

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

We propose a 3-form in differential geometry which depends only of a connection over the tangent fiber bundle.

**Category:** Geometry

[344] **viXra:1903.0241 [pdf]**
*submitted on 2019-03-12 13:24:51*

**Authors:** Abdelmajid Ben Hadj Salem

**Comments:** 10 Pages.

In this note, we give an application of the Method of the Repère Mobile to the Ellipsoid of Reference in Geodesy using a symplectic approach.

**Category:** Geometry

[343] **viXra:1903.0126 [pdf]**
*submitted on 2019-03-07 10:25:27*

**Authors:** Colin James III

**Comments:** 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

From the classical logic section on set theory, we evaluate definitions of the atom and primitive set. None is tautologous. From the quantum logic and topology section on set theory, we evaluate the disjoint union (as equivalent to the XOR operator) and variances in equivalents for the AND and OR operators. None is tautologous. This reiterates that set theory and quantum logic are not bivalent, and hence non-tautologous segments of the universal logic VŁ4. The assertion of Riemannian geometry as generalization of Euclidean geometry is not supported.

**Category:** Geometry

[342] **viXra:1903.0100 [pdf]**
*submitted on 2019-03-07 05:46:58*

**Authors:** Johan Noldus

**Comments:** 67 Pages.

Non commutative geometry is developed from the point of view of an extension of quantum logic. We provide for an example of a non-abelian simplex as well as a non-abelian curved Riemannian space.

**Category:** Geometry

[341] **viXra:1903.0082 [pdf]**
*submitted on 2019-03-05 20:48:15*

**Authors:** Colin James III

**Comments:** 3 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

From the area and dimensions of an outer triangle, the height point of an inner triangle implies the minimum distance to the outer triangle. This proves the solution of Bellman's Lost in the forest problem for triangles. By extension, it is the general solution proof for other figures.

**Category:** Geometry

[340] **viXra:1903.0023 [pdf]**
*submitted on 2019-03-01 09:40:43*

**Authors:** Helmut Söllinger

**Comments:** 10 Pages. language: German

The paper analyses the issue of optimised packaging of spheres of the same size. The question is whether a linear packaging of spheres in the shape of a sausage or a spatial cluster of spheres can minimise the volume enveloping the spheres. There is an assumption that for less than 56 spheres the linear packaging is denser and for 56 spheres the cluster is denser, but the question remains how a cluster of 56 spheres could look like. The paper shows two possible ways to build such a cluster of 56 spheres. The author finds clusters of 59, 62, 65, 66, 68, 69, 71, 72, 74, 75, 76, 77, 78, 79 and 80 spheres - using the same method - which are denser than a linear packaging of the same number and gets to the assumption that all convex clusters of spheres of sufficient size are denser than linear ones.

**Category:** Geometry

[339] **viXra:1902.0444 [pdf]**
*submitted on 2019-02-25 06:00:04*

**Authors:** Madhur Sorout

**Comments:** 12 Pages.

The equivalence of closed figures and infinitely extended lines may lead us to understand the
physical reality of infinities. This paper doesn’t include what infinities mean in the physical
world, but the paper is mainly focused on the equivalence of closed figures and infinitely
extended lines. Using this principle, some major conclusions can be drawn. The equivalence
of closed figures and infinitely extended lines is mainly based on the idea that closed figures
and infinitely extended lines are equivalent. One of the most significant conclusions drawn
from this equivalency is that if any object moves along a straight infinitely extended line, it
will return back to the point, where it started to move, after some definite time.

**Category:** Geometry

[338] **viXra:1902.0401 [pdf]**
*submitted on 2019-02-23 08:07:46*

**Authors:** Eckhard Hitzer

**Comments:** 15 Pages. Submitted to Topical Collection of Adv. in Appl. Clifford Algebras, for Proceedings of AGACSE 2018, 23 Feb. 2019.

This work explains how three dimensional quadrics can be defined by the outer products of conformal geometric algebra points in higher dimensions. These multivector expressions code all types of quadrics in arbitrary scale, location and orientation. Furthermore a newly modified (compared to Breuils et al, 2018, https://doi. org/10.1007/s00006-018-0851-1.) approach now allows not only the use of the standard intersection operations, but also of versor operators (scaling, rotation, translation). The new algebraic form of the theory will be explained in detail.

**Category:** Geometry

[337] **viXra:1902.0370 [pdf]**
*submitted on 2019-02-21 09:48:48*

**Authors:** Antoine Balan

**Comments:** 1 page, written in english

We define the notion of G-connections over vector fiber bundles with action of a Lie group G.

**Category:** Geometry

[336] **viXra:1902.0283 [pdf]**
*submitted on 2019-02-16 15:24:00*

**Authors:** Antoine Balan

**Comments:** 1 page, written in english

We construct a symplectic Laplacian which is a differential operator of order 1 depending only on a connection and a symplectic form.

**Category:** Geometry

[335] **viXra:1902.0028 [pdf]**
*submitted on 2019-02-02 12:41:24*

**Authors:** Antoine Balan

**Comments:** 1 page, written in english

We propose a generalization of the Clifford algebra. We give application to the Dirac operator.

**Category:** Geometry

[334] **viXra:1901.0471 [pdf]**
*submitted on 2019-01-31 08:47:29*

**Authors:** Alexander Skutin

**Comments:** 5 Pages.

In this short note we introduce the blow-up of the Feuerbach’s theorem.

**Category:** Geometry

[333] **viXra:1901.0195 [pdf]**
*submitted on 2019-01-14 11:16:25*

**Authors:** Antoine Balan

**Comments:** 1 pages, written in english

We define a closed 2-form for any spinorial manifold. We deduce characteristic classes.

**Category:** Geometry

[332] **viXra:1901.0162 [pdf]**
*submitted on 2019-01-11 17:27:46*

**Authors:** Hiroshi Okumura

**Comments:** 3 Pages.

We generalize several Archimedean circles, which are the incircles of special triangles.

**Category:** Geometry

[331] **viXra:1901.0152 [pdf]**
*submitted on 2019-01-11 06:31:41*

**Authors:** Edgar Valdebenito

**Comments:** 63 Pages.

Esta nota muestra una colección de fractales.

**Category:** Geometry

[330] **viXra:1812.0226 [pdf]**
*submitted on 2018-12-12 06:35:09*

**Authors:** Edgar Valdebenito

**Comments:** 108 Pages.

This note presents a collection of elementary fractals.

**Category:** Geometry

[329] **viXra:1812.0206 [pdf]**
*submitted on 2018-12-11 21:37:57*

**Authors:** James A. Smith

**Comments:** 5 Pages.

Because the shortage of worked-out examples at introductory levels is an obstacle to widespread adoption of Geometric Algebra (GA), we use GA to solve one of the beautiful \emph{sangaku} problems from 19th-Century Japan. Among the GA operations that prove useful is the rotation of vectors via the unit bivector i.

**Category:** Geometry

[328] **viXra:1812.0090 [pdf]**
*submitted on 2018-12-05 19:29:23*

**Authors:** Colin James III

**Comments:** 3 Pages. © Copyright 2016-2018 by Colin James III All rights reserved. Updated abstract at ersatz-systems.com . Respond to the author by email at: info@ersatz-systems dot com.

We evaluate the axioms of the title. The axiom of identity of betweenness and axiom Euclid are tautologous, but the others are not. The commonplace expression of the axiom of Euclid does not match its other two variations which is troubling. This effectively refutes the planar R-geometry.

**Category:** Geometry

[327] **viXra:1812.0085 [pdf]**
*submitted on 2018-12-04 07:19:40*

**Authors:** Hannes Hutzelmeyer

**Comments:** 93 Pages.

Geometries of O adhere to Ockham's principle of simplest possible ontology: the only individuals are points, there are no straight lines, circles, angles etc. , just as it was was laid down by Tarski in the 1920s, when he put forward a set of axioms that only contain two relations, quaternary congruence and ternary betweenness. However, relations are not as intuitive as functions when constructions are concerned. Therefore the planar geometries of O contain only functions and no relations to start with. Essentially three quaternary functions occur: appension for line-joining of two pairs of points, linisection representing intersection of straight lines and circulation corresponding to intersection of circles. Functions are strictly defined by composition of given ones only. Both, Euclid and Lobachevsky planar geometries are developed using a precise notation for object-language and metalanguage, that allows for a very broad area of mathematical systems up to theory of types. Some astonishing results are obtained, among them: (A) Based on a special triangle construction Euclid planar geometry can start with a less powerful ontological basis than Lobachevsky geometry. (B) Usual Lobachevsky planar geometry is not complete, there are nonstandard planar Lobachevsky geometries. One needs a further axiom, the 'smallest' system is produced by the proto-octomidial-axiom. (C) Real numbers can be abandoned in connection with planar geometry. A very promising conjecture is put forward stating that the Euclidean Klein-model of Lobachevsky planar geometry does not contain all points of the constructive Euclidean unit-circle.

**Category:** Geometry

[326] **viXra:1812.0061 [pdf]**
*submitted on 2018-12-03 06:42:06*

**Authors:** Edgar Valdebenito

**Comments:** 119 Pages.

This note presents a collection of elementary Fractals.

**Category:** Geometry

[325] **viXra:1811.0435 [pdf]**
*submitted on 2018-11-26 06:16:16*

**Authors:** Edgar Valdebenito

**Comments:** 101 Pages.

This note presents a collection of elementary Fractals.

**Category:** Geometry

[324] **viXra:1811.0324 [pdf]**
*submitted on 2018-11-20 06:38:27*

**Authors:** Edgar Valdebenito

**Comments:** 113 Pages.

This note presents a collection of elementary Fractals.

**Category:** Geometry

[323] **viXra:1811.0214 [pdf]**
*submitted on 2018-11-13 06:40:17*

**Authors:** Edgar Valdebenito

**Comments:** 109 Pages.

This note presents a collection of elementary fractals.

**Category:** Geometry

[322] **viXra:1811.0132 [pdf]**
*submitted on 2018-11-08 20:40:58*

**Authors:** Hiroshi Okumura

**Comments:** 5 Pages.

We generalize two sangaku problems involving an arbelos proposed by Izumiya and Nait\=o, and show the existence of six non-Archimedean congruent circles.

**Category:** Geometry

[321] **viXra:1811.0103 [pdf]**
*submitted on 2018-11-06 09:20:05*

**Authors:** Adham Ahmed Mohamed Ahmed

**Comments:** 1 Page.

this paper talks about a hypothesis between the cube and the sphere which is inside the cube and the excess volume of the cube than the sphere and the excess volume of the sphere where the cube is inside of the sphere
What If you spin a cube around an axis passing through its midpoint of the cube would the cylinder formed have an excess in volume than the sphere equal to the excess in volume of the cylinder than he cube?

**Category:** Geometry

[320] **viXra:1810.0379 [pdf]**
*submitted on 2018-10-24 03:13:11*

**Authors:** Франц Герман

**Comments:** 9 Pages.

Что такое след проективной плоскости и как можно его увидеть рассказывается в этой заметке.

**Category:** Geometry

[319] **viXra:1810.0324 [pdf]**
*submitted on 2018-10-21 03:08:17*

**Authors:** Hongbing Zhang

**Comments:** 18 Pages. Please Indicate This Source From Hongbing Zhang When Cite the Contents in Works of Sience or Popular Sience

Why does a half-angle-rotation in quaternion space or spin space correspond to a whole-angle-rotation in normal 3D space? The question is equivalent to why a half angle in the representation of SU(2) corresponds to a whole angle in the representation of SO(3). Usually we use the computation of the abstract mathematics to answer the question. But now I will give an exact and intuitive geometry-explanation on it in this paper.

**Category:** Geometry

[318] **viXra:1810.0295 [pdf]**
*submitted on 2018-10-18 09:47:40*

**Authors:** Франц Герман

**Comments:** Pages.

В данной заметке мы покажем представление дельта-функции Дирака, которое будем назвать естественным. Существующие способы представления дельта-функции Дирака носят в общем-то искусственный характер.

**Category:** Geometry

[317] **viXra:1810.0283 [pdf]**
*submitted on 2018-10-17 05:53:48*

**Authors:** Jan Hakenberg

**Comments:** 10 Pages.

Geodesic averages have been used to generalize curve subdivision and Bézier curves to Riemannian manifolds and Lie groups. We show that geodesic averages are suitable to perform smoothing of sequences of data in nonlinear spaces. In applications that produce temporal uniformly sampled manifold data, the smoothing removes high-frequency components from the signal. As a consequence, discrete differences computed from the smoothed sequence are more regular. Our method is therefore a simpler alternative to the extended Kalman filter. We apply the smoothing technique to noisy localization estimates of mobile robots.

**Category:** Geometry

[316] **viXra:1810.0171 [pdf]**
*submitted on 2018-10-10 13:28:08*

**Authors:** Antoine Balan

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

By similarity with the Seiberg-Witten equations, we propose a set of two equations, depending of a spinor and a vector field.

**Category:** Geometry

[315] **viXra:1810.0068 [pdf]**
*submitted on 2018-10-06 02:12:13*

**Authors:** Johan Noldus

**Comments:** 11 Pages.

We introduce the reader to the problematic aspects of formulating in concreto a suitable notion of geometry. Here, we take the canonical approach and give some examples.

**Category:** Geometry

[314] **viXra:1810.0057 [pdf]**
*submitted on 2018-10-04 09:55:33*

**Authors:** Франц Герман

**Comments:** Pages.

Сформулирована и доказана теорема, ранее не встречавшаяся в литературе по проективной геометрии.
На основании «теоремы о поляре трёхвершинника» открывается целый класс задач на построение.
Теорема может быть полезна студентам математических факультетов педагогических вузов, а также учителям математики средней школы для проведения факультативных занятий.

**Category:** Geometry

[313] **viXra:1809.0515 [pdf]**
*submitted on 2018-09-24 07:45:56*

**Authors:** Edgar Valdebenito

**Comments:** 15 Pages.

En esta nota mostramos algunos fractales del tipo Newton asociados al polinomio: p(z)=z^9+3z^6+3z^3-1,z complejo.

**Category:** Geometry

[312] **viXra:1809.0472 [pdf]**
*submitted on 2018-09-22 12:54:37*

**Authors:** Antoine Balan

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

We define here a generalization of the well-know Levi-Civita connection. We choose an automorphism and define a connection with help of a (non-symmetric) bilinear form.

**Category:** Geometry

[311] **viXra:1809.0323 [pdf]**
*submitted on 2018-09-15 09:41:12*

**Authors:** Antoine Balan

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

We propose here a generalization of the Clifford algebra by mean of two endomorphisms. We deduce a generalized Lichnerowicz formula for the space of modified spinors.

**Category:** Geometry

[310] **viXra:1808.0595 [pdf]**
*submitted on 2018-08-25 08:47:43*

**Authors:** Somsikov A.I.

**Comments:** 2 Pages.

One of initial or primary concepts which is considered to the "protozoa" (who aren't expressed through other concepts) is considered. The structure of this concept is revealed. Algebraic and geometrical consequences are found.
Рассмотрено одно из исходных или первичных понятий, считающееся «простейшим» (не выражаемым через другие понятия). Выявлена структура этого понятия. Найдены алгебраические и геометрические следствия.

**Category:** Geometry

[309] **viXra:1808.0208 [pdf]**
*submitted on 2018-08-15 11:10:29*

**Authors:** Andrei Lucian Dragoi

**Comments:** 15 Pages.

This paper brings to attention the intrinsic paradox of the geometric point (GP) definition, a paradox solved in this paper by using Stéphane Lupasco’s Included Middle Logic (IML) (which was stated by Basarab Nicolescu as one of the three pillars of transdisciplinarity [TD]) and its extended form: based on IML, a new “t-metamathematics” (tMM) (including a t-metageometry[tMG]) is proposed, which may explain the main cause of Euclid’s parallel postulate (EPP) “inaccuracy”, allowing the existence not only of non-Euclidean geometries (nEGs), but also the existence of new EPP variants. tMM has far-reaching implications, including the help in redefining the basics of Einstein’s General relativity theory (GRT), quantum field theory (QFT), superstring theories (SSTs) and M-theory (MT).
KEYWORDS (including a list of main abbreviations): geometric point (GP); Stéphane Lupasco’s Included Middle Logic (IML); Basarab Nicolescu, transdisciplinarity (TD); “t-metamathematics” (tMM); t-metageometry (tMG); Euclid’s parallel postulate (EPP); non-Euclidean geometries; new EPP variants; Einstein’s General relativity theory (GRT); quantum field theory (QFT); superstring theories (SSTs); M-theory (MT);

**Category:** Geometry

[308] **viXra:1808.0206 [pdf]**
*submitted on 2018-08-15 12:34:49*

**Authors:** Antoine Balan

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

We define here the notion of Balan-Killing manifolds which are solutions of differential equations over the metrics of spin manifolds.

**Category:** Geometry

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

**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

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

**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

[305] **viXra:1807.0234 [pdf]**
*submitted on 2018-07-12 16:30:22*

**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

[304] **viXra:1806.0116 [pdf]**
*submitted on 2018-06-09 15:36:41*

**Authors:** Antoine Balan

**Comments:** 1 page, written in english

In the case of a manifold which is a Lie group, a Dirac operator can be defined acting over the vector fields of the Lie group instead of the spinors.

**Category:** Geometry

[303] **viXra:1805.0030 [pdf]**
*submitted on 2018-05-01 00:29:25*

**Authors:** Johan Aspegren

**Comments:** 7 Pages.

One theme of this paper is to extend known results from polygons and balls to the general convex bodies in n− dimensions. An another theme stems from approximating a convex surface with polytope surface. Our result gives a sufficient and necessary condition for an natural approximation method to succeed (in principle) in the case of surfaces of convex bodies. Thus, Schwartz`s paradox does not affect our method. This allows us to denefine certain surface measures on surfaces of convex bodies in a novel and simple way.

**Category:** Geometry

[302] **viXra:1804.0397 [pdf]**
*submitted on 2018-04-27 03:22:51*

**Authors:** Zhenghan Shen, Wen Wang, Pan Zhang

**Comments:** 9 Pages.

In this paper, by the method of heat flow and the
method of exhaustion, we prove an existence theorem of Hermitian-Yang-Mills-Higgs metrics on holomorphic line bundle over a class of non-compact Gauduchon manifold.

**Category:** Geometry

[301] **viXra:1804.0363 [pdf]**
*submitted on 2018-04-24 20:34:14*

**Authors:** James A. Smith

**Comments:** 54 Pages.

Because the shortage of worked-out examples at introductory levels is an obstacle to widespread adoption of Geometric Algebra (GA), we use GA to calculate Solar azimuths and altitudes as a function of time via the heliocentric model. We begin by representing the Earth's motions in GA terms. Our representation incorporates an estimate of the time at which the Earth would have reached perihelion in 2017 if not affected by the Moon's gravity. Using the geometry of the December 2016 solstice as a starting point, we then employ GA's capacities for handling rotations to determine the orientation of a gnomon at any given latitude and longitude during the period between the December solstices of 2016 and 2017. Subsequently, we derive equations for two angles: that between the Sun's rays and the gnomon's shaft, and that between the gnomon's shadow and the direction ``north" as traced on the ground at the gnomon's location. To validate our equations, we convert those angles to Solar azimuths and altitudes for comparison with simulations made by the program Stellarium. As further validation, we analyze our equations algebraically to predict (for example) the precise timings and locations of sunrises, sunsets, and Solar zeniths on the solstices and equinoxes. We emphasize that the accuracy of the results is only to be expected, given the high accuracy of the heliocentric model itself, and that the relevance of this work is the efficiency with which that model can be implemented via GA for teaching at the introductory level. On that point, comments and debate are encouraged and welcome.

**Category:** Geometry

[300] **viXra:1804.0360 [pdf]**
*submitted on 2018-04-25 02:11:06*

**Authors:** Hiroshi Okumura

**Comments:** 2 Pages. This paper will be submitted to Sangaku Journal of Mathematics.

A problem involving an isosceles triangle with a square and three congruent circles is generalized.

**Category:** Geometry

[120] **viXra:1910.0620 [pdf]**
*replaced on 2019-10-31 18:02:58*

**Authors:** Dante Servi

**Comments:** 10 Pages.

Further reflections related to my article published on viXra.org in the geometry group at number 1910.0086 (revision v3) File name: 1910.0086v3.pdf with the following title: Poligonal spirals with manageable inclination complete version of the discussion. ---- Note: My article to which this refers is found below, at number 1910.0086.
As is for my article to which this refers, it is possible that also this article of mine is updated, to be sure to download the latest revision do not click on (pdf) but on (viXra: nnnn.nnnn), it will open the page where all the revisions are, (v1), (v2), (v3), (v...). On this page click on (v...) to download the latest revision.

**Category:** Geometry

[119] **viXra:1910.0185 [pdf]**
*replaced on 2019-10-15 07:04:59*

**Authors:** Antoine Balan

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

In the realm of the riemannian geometry, a moduli space is defined.

**Category:** Geometry

[118] **viXra:1910.0185 [pdf]**
*replaced on 2019-10-14 05:57:54*

**Authors:** Antoine Balan

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

A moduli space of vector fields is defined for any riemannian manifold.

**Category:** Geometry

[117] **viXra:1910.0086 [pdf]**
*replaced on 2019-10-15 15:44:38*

**Authors:** Dante Servi

**Comments:** 29 Pages.

Descrizione di un tipo di spirale composto da un insieme di segmenti che con riferimento ad un punto che definisco origine hanno una inclinazione gestibile e volendo costante. Descrizione di metodo grafico e di algoritmi che permettono di realizzarlo.
Nel foglio 10/10 descrivo come realizzare una spirale poligonale che ha tutti i vertici in comune con una spirale logaritmica.
Nel foglio 10 bis descrivo come calcolare (dopo aver deciso il grado di precisione con cui si vuol seguire il percorso della logaritmica) l'inclinazione da attribuire ai segmenti destinati a realizzare la spirale poligonale.
Sempre nel foglio 10 bis affermo che il mio metodo utilizzato al contrario può essere almeno provato per studiare in un modo nuovo una curva sconosciuta. Di seguito provo a confrontare il mio metodo con la spirale di Archimede, ricavando le informazioni utili per realizzare una poligonale che abbia tutti i suoi vertici in comune con essa, sia graficamente che definendo un algoritmo.
Description of a type of spiral composed of a set of segments that with a point that I define origin have a manageable inclination and wanting to be constant. Description of graphic method and algorithms that allow to realize it.
In sheet 10/10 I describe how to make a polygonal spiral that has all the vertices in common with a logarithmic spiral.
In sheet 10 bis I describe how to calculate (after deciding the degree of precision with which we want to follow the path of the logarithmic) the inclination to be attributed to the segments destined to realize the polygonal spiral.
Also in sheet 10 bis I state that my method used on the contrary can at least be tried to study an unknown curve in a new way. Next I try to compare my method with the Archimede spiral, obtaining the information useful for creating a polygon that has all its vertices in common with it, both graphically and by defining an algorithm.

**Category:** Geometry

[116] **viXra:1910.0086 [pdf]**
*replaced on 2019-10-12 07:39:18*

**Authors:** Dante Servi

**Comments:** 29 Pages.

Descrizione di un tipo di spirale composto da un insieme di segmenti che con riferimento ad un punto che definisco origine hanno una inclinazione gestibile e volendo costante. Descrizione di metodo grafico e di algoritmi che permettono di realizzarlo.
Nel foglio 10/10 descrivo come realizzare una spirale poligonale che ha tutti i vertici in comune con una spirale logaritmica.
Nel foglio 10 bis descrivo come calcolare (dopo aver deciso il grado di precisione con cui si vuol seguire il percorso della logaritmica) l'inclinazione da attribuire ai segmenti destinati a realizzare la spirale poligonale.
Sempre nel foglio 10 bis affermo che il mio metodo utilizzato al contrario può essere almeno provato per studiare in un modo nuovo una curva sconosciuta. Io non posso affermare di aver studiato con il mio metodo la spirale logaritmica però ne ho ricavato l'algoritmo illustrato nel foglio 10/10.
Description of a type of spiral composed of a set of segments that with a point that I define origin have a manageable inclination and wanting to be constant. Description of graphic method and algorithms that allow to realize it.
In sheet 10/10 I describe how to make a polygonal spiral that has all the vertices in common with a logarithmic spiral.
In sheet 10 bis I describe how to calculate (after deciding the degree of precision with which we want to follow the path of the logarithmic) the inclination to be attributed to the segments destined to realize the polygonal spiral.
Also in sheet 10 bis I state that my method used on the contrary can at least be tried to study an unknown curve in a new way. I cannot claim to have studied the logarithmic spiral with my method, but I have derived the algorithm illustrated in sheet 10/10.

**Category:** Geometry

[115] **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

[114] **viXra:1908.0074 [pdf]**
*replaced on 2019-08-12 04:48:42*

**Authors:** Atsushi Koike

**Comments:** 14 Pages.

According to Pierre Wantzel’s proof of 1837 that the trisection of 60 degree is impossible, because the cubic equation of x^3 - 3x - 1 = 0 had a absence of a rational solution. And his proof reached already a consensus as a general opinion. I learned from Nobukazu Shimeno’s introductory book on complex numbers that Rafael Bombelli got a rational solution x= 4 from the cubic equation x^3 = 15x + 4 based on the Cardinal formula. x^3 = 15x + 4 is x^3 - 15x - 4 = 0, which can be further replaced by x^3 - dx - a = 0. Kentaro Yano, who introduces the trisection of angles, says that the basic equation of x^3 - 3x - 1 = 0 is x^3 - dx - a = 0. In other words, the equation that Rafael Bombelli obtained a rational number solution is the same as the equation of the trisection of the angle. On the other hand, Yano raises x^3 - 3x = 0 as an example when the angle can be divided into three equal parts. Needless to say, x^3 - 3x - 0 = 0 and x^3 - dx - a = 0. Therefore, comparing Rafael Bombelli’s solution and the equation where Yano’s angle trisection is impossible and possible, the equation for angle trisection is x = a. And I found that if x = a = 2 then the solution is obtained at all angles. This paper proves that.

**Category:** Geometry

[113] **viXra:1908.0074 [pdf]**
*replaced on 2019-08-08 09:33:02*

**Authors:** Atsushi Koike

**Comments:** 14 Pages.

According to Pierre Wantzel’s proof of 1837 that the trisection of 60 degree is impossible, because the cubic equation of had a absence of a rational solution. And his proof reached already a consensus as a general opinion. I learned from Nobukazu Shimeno’s introductory book on complex numbers that Rafael Bombelli got a rational solution from the cubic equation based on the Cardinal formula. is , which can be further replaced by . Kentaro Yano, who introduces the trisection of angles, says that the basic equation of is . In other words, the equation that Rafael Bombelli obtained a rational number solution is the same as the equation of the trisection of the angle. On the other hand, Yano raises as an example when the angle can be divided into three equal parts. Needless to say, and . Therefore, comparing Rafael Bombelli’s solution and the equation where Yano’s angle trisection is impossible and possible, the equation for angle trisection is . And I found that if then the solution is obtained at all angles. This paper proves that.
Category: Geometry

**Category:** Geometry

[112] **viXra:1907.0581 [pdf]**
*replaced on 2019-08-26 11:18:55*

**Authors:** Hiroshi Okumura

**Comments:** 5 Pages.

We consider a problem in Wasan geometry involving
a golden arbelos and give a characterization of the golden arbelos involving an Archimedean circle. We also construct a self-similar circle configuration using the figure of the problem.

**Category:** Geometry

[111] **viXra:1907.0581 [pdf]**
*replaced on 2019-07-29 22:51:43*

**Authors:** Hiroshi Okumura

**Comments:** 5 Pages.

We consider a problem in Wasan geometry involving a golden arbelos.

**Category:** Geometry

[110] **viXra:1907.0496 [pdf]**
*replaced on 2019-07-25 15:42:08*

**Authors:** Yuji Masuda

**Comments:** 1 Page.

This shows a geometry conposed of i,1,-1,∞.

**Category:** Geometry

[109] **viXra:1905.0030 [pdf]**
*replaced on 2019-09-13 02:27:49*

**Authors:** Eckhard Hitzer, Stephen J. Sangwine

**Comments:** Adv. of App. Cliff. Algs., (2019) 29(5):96 (First Online: 04 October 2019), 16 pages, DOI: 10.1007/s00006-019-1016-6, 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 Hrdina et al. (Adv. Appl Cliff. Algs. Vol. 28:66, pp. 1–21, https://doi.org/10.1007/s00006-018-0879-2,2018), but simpler, especially for translations.

**Category:** Geometry

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

**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

[107] **viXra:1905.0026 [pdf]**
*replaced on 2019-05-03 10:07:54*

**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, correction: 03 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

[106] **viXra:1903.0560 [pdf]**
*replaced on 2019-05-06 19:59:05*

**Authors:** Anamitra Palit

**Comments:** 5 Pages.

The direct sum decomposition of a vector space has been explored to bring out a conflicting feature in the theory. We decompose a vector space using two subspaces. Keeping one subspace fixed we endeavor to replace the other by one which is not equal to the replaced subspace. Proceeding from such an effort we bring out the conflict. From certain considerations it is not possible to work out the replacement with an unequal subspace. From alternative considerations an unequal replacement is possible.

**Category:** Geometry

[105] **viXra:1903.0317 [pdf]**
*replaced on 2019-03-18 18:09:25*

**Authors:** Colin James III

**Comments:** 2 Pages.

We prove two parallel lines are tautologous in Euclidean geometry. We next prove that non Euclidean geometry of Lobachevskii is not tautologous and hence not consistent. What follows is that Riemann geometry is the same, and non Euclidean geometry is a segment of Euclidean geometry, not the other way around. Therefore non Euclidean geometries are a non tautologous fragment of the universal logic VŁ4.

**Category:** Geometry

[104] **viXra:1902.0444 [pdf]**
*replaced on 2019-02-26 08:55:44*

**Authors:** Madhur Sorout

**Comments:** 13 Pages.

This paper is mainly focused on the equivalence of closed figures and infinitely extended
lines. Using this principle, some major conclusions can be drawn. The equivalence of closed
figures and infinitely extended lines is mainly based on the idea that closed figures and
infinitely extended lines are equivalent. One of the most significant conclusions drawn from
this equivalency is that if any object moves along a straight infinitely extended line, it will
return back to the point, where it started to move, after some definite time. This principle of
equivalence of closed figures and infinitely extended lines may lead us to understand the
physical reality of infinities.

**Category:** Geometry

[103] **viXra:1902.0401 [pdf]**
*replaced on 2019-04-16 08:32:36*

**Authors:** Eckhard Hitzer

**Comments:** 16 Pages. published in Adv. of App. Cliff. Algs., 29:46, pp. 1-16, 2019. DOI: 10.1007/s00006-019-0964-1, 1 table.

This work explains how three dimensional quadrics can be defined by the outer products of conformal geometric algebra points in higher dimensions. These multivector expressions code all types of quadrics in arbitrary scale, location and orientation. Furthermore, a newly modified (compared to Breuils et al, 2018, https://doi.org/10.1007/s00006-018-0851-1.) approach now allows not only the use of the standard intersection operations, but also of versor operators (scaling, rotation, translation). The new algebraic form of the theory will be explained in detail.

**Category:** Geometry

[102] **viXra:1902.0401 [pdf]**
*replaced on 2019-03-02 02:42:11*

**Authors:** Eckhard Hitzer

**Comments:** Submitted to Topical Collection of Adv. in Appl. Clifford Algebras, for Proceedings of AGACSE 2018, 23 Feb. 2019, 15 pages. 4 errors corrected: 25 Feb. 2019. Proposition 4.1 corrected: 02 Mar. 2019.

This work explains how three dimensional quadrics can be defined by
the outer products of conformal geometric algebra points in higher dimensions.
These multivector expressions code all types of quadrics in arbitrary scale, location
and orientation. Furthermore, a newly modified (compared to Breuils et al, 2018, https://doi.org/10.1007/s00006-018-0851-1.) approach
now allows not only the use of the standard intersection operations, but also of
versor operators (scaling, rotation, translation). The new algebraic form of the
theory will be explained in detail.

**Category:** Geometry

[101] **viXra:1902.0401 [pdf]**
*replaced on 2019-02-25 06:01:09*

**Authors:** Eckhard Hitzer

**Comments:** Submitted to Topical Collection of Adv. in Appl. Clifford Algebras, for Proceedings of AGACSE 2018, 23 Feb. 2019, 15 pages. 4 errors corrected: 25 Feb. 2019.

This work explains how three dimensional quadrics can be defined by the outer products of conformal geometric algebra points in higher dimensions. These multivector expressions code all types of quadrics in arbitrary scale, location and orientation. Furthermore a newly modified (compared to Breuils et al, 2018, https://doi.org/10.1007/s00006-018-0851-1.) approach now allows not only the use of the standard intersection operations, but also of versor operators (scaling, rotation, translation). The new algebraic form of the theory will be explained in detail.

**Category:** Geometry

[100] **viXra:1812.0423 [pdf]**
*replaced on 2019-10-31 14:20:25*

**Authors:** Shawn Halayka

**Comments:** 9 Pages.

In this short memorandum, the curvature and dimension properties of the $2$-sphere surface of a 3-dimensional ball and the $2.x$-dimensional surface of a 3-dimensional fractal set are considered.
Tessellation is used to approximate each surface, primarily because the $2.x$-dimensional surface of a 3-dimensional fractal set is otherwise non-differentiable (having no well-defined surface normals).
It is found that the curvature of a closed surface {\it must} lead to fractional dimension.

**Category:** Geometry

[99] **viXra:1812.0423 [pdf]**
*replaced on 2019-10-22 21:39:38*

**Authors:** Shawn Halayka

**Comments:** 8 Pages.

In this short memorandum, the curvature and dimension properties of the $2$-sphere surface of a 3-dimensional ball and the $2.x$-dimensional surface of a 3-dimensional fractal set are considered.
Tessellation is used to approximate each surface, primarily because the $2.x$-dimensional surface of a 3-dimensional fractal set is otherwise non-differentiable (having no well-defined surface normals).
It is found that the curvature of a closed surface {\it must} lead to fractional dimension.

**Category:** Geometry

[98] **viXra:1812.0423 [pdf]**
*replaced on 2019-09-16 18:42:23*

**Authors:** Shawn Halayka

**Comments:** 5 Pages.

The curvature of a surface can lead to fractional dimension.
In this paper, the properties of the 2-sphere surface of a 3D ball and the 2.x-surface of a 3D fractal set are considered.
Tessellation is used to approximate each surface, primarily because the 2.x-surface of a 3D fractal set is otherwise non-differentiable.

**Category:** Geometry

[97] **viXra:1812.0423 [pdf]**
*replaced on 2019-01-11 20:21:50*

**Authors:** Shawn Halayka

**Comments:** 5 Pages.

The curvature of a surface can lead to fractional dimension.
In this paper, the properties of the 2-sphere surface of a 3D ball and the 2.x-surface of a 3D fractal set are considered.
Tessellation is used to approximate each surface, primarily because the 2.x-surface of a 3D fractal set is otherwise non-differentiable.

**Category:** Geometry

[96] **viXra:1812.0423 [pdf]**
*replaced on 2019-01-08 10:44:26*

**Authors:** Shawn Halayka

**Comments:** 5 Pages.

The curvature of a surface can lead to fractional dimension.
In this paper, the properties of the 2-sphere surface of a 3D ball and the 2.x-surface of a 3D fractal set are considered.
Tessellation is used to approximate each surface, primarily because the 2.x-surface of a 3D fractal set is otherwise non-differentiable.

**Category:** Geometry

[95] **viXra:1812.0206 [pdf]**
*replaced on 2019-07-26 18:11:40*

**Authors:** James A. Smith

**Comments:** 8 Pages.

Because the shortage of worked-out examples at introductory levels is an obstacle to widespread adoption of Geometric Algebra (GA), we use GA to solve one of the beautiful sangaku problems from 19th-Century Japan. Among the GA operations that prove useful is the rotation of vectors via the unit bivector, i.

**Category:** Geometry

[94] **viXra:1812.0085 [pdf]**
*replaced on 2019-01-24 08:52:30*

**Authors:** Hannes Hutzelmeyer

**Comments:** 93 Pages.

Geometries of O adhere to Ockham's principle of simplest possible ontology: the only individuals are points, there are no straight lines, circles, angles etc. , just as it was was laid down by Tarski in the 1920s, when he put forward a set of axioms that only contain two relations, quaternary congruence and ternary betweenness. However, relations are not as intuitive as functions when constructions are concerned. Therefore the planar geometries of O contain only functions and no relations to start with. Essentially three quaternary functions occur: appension for line-joining of two pairs of points, linisection representing intersection of straight lines and circulation corresponding to intersection of circles. Functions are strictly defined by composition of given ones only. Both, Euclid and Lobachevsky planar geometries are developed using a precise notation for object-language and metalanguage, that allows for a very broad area of mathematical systems up to theory of types. Some astonishing results are obtained, among them: (A) Based on a special triangle construction Euclid planar geometry can start with a less powerful ontological basis than Lobachevsky geometry. (B) Usual Lobachevsky planar geometry is not complete, there are nonstandard planar Lobachevsky geometries. One needs a further axiom, the 'smallest' system is produced by the proto-octomidial- axiom. (C) Real numbers can be abandoned in connection with planar geometry. A very promising conjecture is put forward stating that the Euclidean Klein-model of Lobachevsky planar geometry does not contain all points of the constructive Euclidean unit-circle.

**Category:** Geometry

[93] **viXra:1810.0324 [pdf]**
*replaced on 2018-10-23 10:40:48*

**Authors:** Hongbing Zhang

**Comments:** 18 Pages. Please Indicate This Source From Hongbing Zhang When Cite the Contents in Works of Sience or Popular Sience

Why does a half-angle-rotation in quaternion space or spin space correspond to a whole-angle-rotation in normal 3D space? The question is equivalent to why a half angle in the representation of SU(2) corresponds to a whole angle in the representation of SO(3). Usually we use the computation of the abstract mathematics to answer the question. But now I will give an exact and intuitive geometry-explanation on it in this paper.

**Category:** Geometry

[92] **viXra:1810.0324 [pdf]**
*replaced on 2018-10-22 05:26:02*

**Authors:** Hongbing Zhang

**Comments:** 18 Pages. Please Indicate This Source From Hongbing Zhang When Cite the Contents in Works of Sience or Popular Sience

Why does a half-angle-rotation in quaternion space or spin space correspond to a whole-angle-rotation in normal 3D space? The question is equivalent to why a half angle in the representation of SU(2) corresponds to a whole angle in the representation of SO(3). Usually we use the computation of the abstract mathematics to answer the question. But now I will give an exact and intuitive geometry-explanation on it in this paper.

**Category:** Geometry

[91] **viXra:1810.0171 [pdf]**
*replaced on 2018-10-14 11:00:59*

**Authors:** Antoine Balan

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

By similarity with the Seiberg-Witten equations, we propose two differential equations, depending of a spinor and a vector field, instead of a connection. Good moduli spaces are espected as a consequence of commutativity.

**Category:** Geometry

[90] **viXra:1810.0171 [pdf]**
*replaced on 2018-10-13 14:13:21*

**Authors:** Antoine Balan

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

By similarity with the Seiberg-Witten equations, we propose two differential equations, depending of a spinor and a vector field, instead of a connection. Good moduli spaces are espected as a consequence of commutativity.

**Category:** Geometry

[89] **viXra:1808.0206 [pdf]**
*replaced on 2018-08-18 16:33:06*

**Authors:** Antoine Balan

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

We define here the notion of Balan-Killing manifolds which are spin manifolds whose metrics verify a certain differential equation. We take our inspiration from the notion of Killing spinors.

**Category:** Geometry

[88] **viXra:1808.0206 [pdf]**
*replaced on 2018-08-18 05:09:12*

**Authors:** Antoine Balan

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

We define here the notion of Balan-Killing manifolds which are spin manifolds whose metrics verify a certain differential equation. We take our inspiration from the notion of Killing spinors.

**Category:** Geometry

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

**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

[86] **viXra:1805.0030 [pdf]**
*replaced on 2019-09-21 11:22:17*

**Authors:** Johan Aspegren

**Comments:** 8 Pages.

One theme of this paper is to extend known results from polygons and balls to the general convex bodies in n− dimensions. An another theme stems from approximating a convex surface with polytope surface. Our result gives a sufficient and necessary condition for an natural approximation method to succeed (in principle) in the case of surfaces of convex bodies. Thus, Schwartz`s paradox does not affect our method. This allows us to denefine certain surface measures on surfaces of convex bodies in a novel and simple way.

**Category:** Geometry

[85] **viXra:1805.0030 [pdf]**
*replaced on 2019-05-30 19:33:17*

**Authors:** Johan Aspegren

**Comments:** 7 Pages.

One theme of this paper is to extend known results from polygons and balls to the general convex bodies in n− dimensions. An another theme stems from approximating a convex surface with polytope surface. Our result gives a sufficient and necessary condition for an natural approximation method to succeed (in principle) in the case of surfaces of convex bodies. Thus, Schwartz`s paradox does not affect our method. This allows us to denefine certain surface measures on surfaces of convex bodies in a novel and simple way.

**Category:** Geometry

[84] **viXra:1805.0030 [pdf]**
*replaced on 2018-05-30 16:22:45*

**Authors:** Johan Aspegren

**Comments:** 7 Pages.

**Category:** Geometry

[83] **viXra:1805.0030 [pdf]**
*replaced on 2018-05-22 11:28:23*

**Authors:** Johan Aspegren

**Comments:** 7 Pages.

One theme of this paper is to extend known results from polygons and balls to the general convex bodies in n− dimensions. An another theme stems from approximating a convex surface with a polytope surface. Our result gives a sufficient and necessary condition for an natural approximation method to succeed (in principle) in the case of surfaces of convex bodies. Thus, Schwartz`s paradox does not affect our method. This allows us to define certain surface measures on surfaces of convex bodies in a novel and simple way.

**Category:** Geometry

[82] **viXra:1805.0030 [pdf]**
*replaced on 2018-05-20 22:09:51*

**Authors:** Johan Aspegren

**Comments:** 7 Pages.

One theme of this paper is to extend known results from polygons and balls to the general convex bodies in n− dimensions. An another theme stems from approximating a convex surface with a polytope surface. Our result gives a sufficient and necessary condition for an natural approximation method to succeed (in principle) in the case of surfaces of convex bodies. Thus, Schwartz`s paradox does not affect our method. This allows us to define certain surface measures on surfaces of convex bodies in a novel and simple way.

**Category:** Geometry

[81] **viXra:1805.0030 [pdf]**
*replaced on 2018-05-06 01:48:26*

**Authors:** Johan Aspegren

**Comments:** 7 Pages.

One theme of this paper is to extend known results from polygons and balls to the general convex bodies in n− dimensions. An another theme stems from approximating a convex surface with a polytope surface. Our result gives a sufficient and necessary condition for an natural approximation method to succeed (in principle) in the case of surfaces of convex bodies. Thus, Schwartz`s paradox does not affect our method. This allows us to define certain surface measures on surfaces of convex bodies in a novel and simple way.

**Category:** Geometry