Classical Geometries in Modern Contexts

Benz, Walter

doi:10.1007/978-3-0348-0420-2

Cited by 13 publications

(14 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…where the product of vectors in E n+1 is understood to be the inner product with respect to the Euclidean metric, is called power of c1 and c2. It turns out that c1 and c2 touch each other respecting orientation if, and only if, (Benz (2012), proposition 3.43) P (c1, c2) = 0, and this is exactly the reflexive and symmetric contact relation studied in sphere geometry. Moreover, V can be equipped with a product defined as c1c2 = −r1r2 + a1a2.…”

Section: Appendixmentioning

confidence: 86%

On de-Sitter geometry in crater statistics

Gibbons¹,

Werner²

2012

Monthly Notices of the Royal Astronomical Society

View full text Add to dashboard Cite

The cumulative size-frequency distributions of impact craters on planetary bodies in the solar system appear to approximate a universal inverse square power-law for small crater radii. In this article, we show how this distribution can be understood easily in terms of geometrical statistics, using a de-Sitter geometry of the configuration space of circles on the Euclidean plane and on the unit sphere. The effect of crater overlap is also considered.

show abstract

Section: Appendixmentioning

confidence: 86%

On de-Sitter geometry in crater statistics

Gibbons¹,

Werner²

2012

Monthly Notices of the Royal Astronomical Society

View full text Add to dashboard Cite

show abstract

“…All results mentioned in this paper belong to a young and active geometrical discipline called "characterizations of geometrical mappings under mild hypotheses". The discipline started around 1950 with fundamental theorems of A. D. Alexandrov on spacetime transformations and causal automorphisms (see [3]). Throughout the conditions in these theorems, for examples, Theorems A ∼ D and our main result, surjectivity, injectivity, hyperplane-preserving and non-degenerate assumptions play inevitable roles.…”

Section: Discussionmentioning

confidence: 99%

Fundamental theorem of hyperbolic geometry without the injectivity assumption

Yao

2011

Mathematische Nachrichten

View full text Add to dashboard Cite

Let H n be the n-dimensional hyperbolic space. It is well-known that, if f : H n → H n is a bijection that preserves r-dimensional hyperplanes, then f is an isometry. In this paper we make neither injectivity nor r-hyperplane preserving assumptions on f and prove the following result:Suppose that f : H n → H n is a surjective map and maps an r-hyperplane into an r-hyperplane, then f is an isometry.The Euclidean version was obtained by Chubarev and Pinelis in 1999 among other things. Our proof is essentially different from their and the similar problem arising in the spherical case is still open.

show abstract

“…The mathematical formalism employed in the library cycle is based on Clifford algebras and the Fillmore-Springer-Cnops construction (FSCc), which has a long history, see [24, § 1.1], [20, § 4.1], [19], [25, § 4.2], [26], [13, § 4.2]. Compared to a plain analytical treatment [27,11], FSCc is much more efficient and conceptually coherent in dealing with FLT-invariant properties of cycles. Correspondingly, the computer code based on FSCc is easy to write and maintain.…”

Section: Problems and Backgroundmentioning

confidence: 99%

“…The associated geometries span a wide domain, which includes conformal [8, Ch. 9], hypercomplex [9] and Lie sphere geometries [10,11,12]. For these fields the package facilitates:…”

Section: Introductionmentioning

confidence: 99%

MoebInv: C++ libraries for manipulations in non-Euclidean geometry

Kisil¹

2020

SoftwareX

View full text Add to dashboard Cite

The introduced package MoebInv contains two C ++ libraries for symbolic, numeric and graphical manipulations in non-Euclidean geometry. The first library cycle implements basic geometric operations on cycles, which are the zero sets of certain polynomials of degree two. The second library figure operates on ensembles of cycles interconnected by Moebius-invariant relations: orthogonality, tangency, etc. Both libraries work in spaces with any dimension and arbitrary signatures of their metrics. Their essential functionality is accessible in interactive modes from Python/Jupyter shells and a dedicated Graphical User Interface. The latter does not require any coding skills and can be used in education. The package is tested on (and supplied for) various Linux distributions, Windows 10, Mac OS X and several cloud services.

show abstract

Classical Geometries in Modern Contexts

Cited by 13 publications

References 0 publications

On de-Sitter geometry in crater statistics

On de-Sitter geometry in crater statistics

Fundamental theorem of hyperbolic geometry without the injectivity assumption

MoebInv: C++ libraries for manipulations in non-Euclidean geometry

Contact Info

Product

Resources

About