zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Evolutes of hyperbolic plane curves. (English) Zbl 1065.53022
Let 1 3 be the Minkowski 3-space, or pseudo-Euclidean space, i.e. the 3-dimensional vector-space 3 equipped with the scalar product x,y=-x 1 y 1 +x 2 y 2 +x 3 y 3 . The Lorentzian sphere: x,x=-1, a two-sheet-hyperboloid, contains as one of its sheets the surface H + 2 : x,x=-1, x 1 1 which is adopted as the model of the hyperbolic plane and (wrongly) called “hyperbola”. The authors develop the Frenet-Serret-type formula for hyperbolic plane curves and study in particular the following properties of these curves: hyperbolic invariants, osculating pseudo-circles, the hyperbolic evolute and its singularities. The case when a point of the hyperbolic evolute is an ordinary cusp is characterized (theorem 5.3). The last section describes how one can draw the picture of the hyperbolic evolute of a curve on the Poincaré disk: fig. 1 shows a family of Euclidean ellipses and their hyperbolic evolutes. The authors mention – what can be seen clearly in fig. 1 – that the four cusps theorem does not hold in H + 2 (although the four vertex theorem does).
MSC:
53B30Lorentz metrics, indefinite metrics
57R70Critical points and critical submanifolds
53A04Curves in Euclidean space
References:
[1]Bruce, J. W., Giblin, P. J.: Curves and singularities (second edition), Cambridge University Press, Glasgow, 1992
[2]Torii, E.: On curves on the hyperboloid or the pseudo-sphere in Minkowski 3-space, Master thesis of Hokkaido University, 1999 (in Japanese)
[3]O’Neill, B.: Semi-Riemannian Geometry, Academic Press, New York, 1983