different between hyperbolic vs euclidean

English

Alternative forms

• hyperbolick (obsolete)

Pronunciation

• (US) enPR: h?p?rb?l'?k, IPA^(key): /?ha?p??b?l?k/
• Rhymes: -?l?k

Etymology 1

hyperbole +? -ic

Adjective

hyperbolic (comparative more hyperbolic, superlative most hyperbolic)

1. Of or relating to hyperbole.
2. Using hyperbole: exaggerated.

Synonyms

• hyperbolical

Translations

Etymology 2

hyperbola +? -ic

Adjective

hyperbolic (not comparable)

1. Of or pertaining to a hyperbola.
   • 1988, R. F. Leftwich, "Wide-Band Radiation Thermometers", chapter 7 of, David P. DeWitt and Gene D. Nutter, editors, Theory and Practice of Radiation Thermometry, ?ISBN, page 512 [2]:

     In this configuration the on-axis image is produced at the real hyperbolic focus (f_s2) but off-axis performance suffers.

2. Indicates that the specified function is a hyperbolic function rather than a trigonometric function.

   The hyperbolic cosine of zero is one.

3. (mathematics, of a metric space or a geometry) Having negative curvature or sectional curvature.
   • 1998, Katsuhiko Matsuzaki and Masahiko Taniguchi, Hyperbolic Manifolds and Kleinian Groups, 2002 reprint, Oxford, ?ISBN, page 8, proposition 0.10 [3]:

     There is a universal constant ${\displaystyle m_{0}>0}$ such that every hyperbolic surface ${\displaystyle R}$ has an embedded hyperbolic disk with radius greater than ${\displaystyle m_{0}}$.

4. (geometry, topology, of an automorphism) Whose domain has two (possibly ideal) fixed points joined by a line mapped to itself by translation.
   • 2001, A. F. Beardon, "The Geometry of Riemann Surfaces", in, E. Bujalance, A. F. Costa, and E. Martínez, editors, Topics on Riemann Surfaces and Fuchsian Groups, Cambridge, ?ISBN, page 6 [4]:

     A hyperbolic isometry ${\displaystyle f}$ has two (distinct) fixed points on ${\displaystyle \partial {\mathcal {H}}}$.

5. (topology) Of, pertaining to, or in a hyperbolic space (a space having negative curvature or sectional curvature).
   • 2001, A. F. Beardon, "The Geometry of Riemann Surfaces", in, E. Bujalance, A. F. Costa, and E. Martínez, editors, Topics on Riemann Surfaces and Fuchsian Groups, Cambridge, ?ISBN, page 6 [5]:

     Exactly one hypercycle is a hyperbolic geodesic, and this is called the axis ${\displaystyle A_{f}}$ of ${\displaystyle f}$.

Derived terms

Translations

English

Adjective

euclidean (not comparable)

1. (rare) Alternative spelling of Euclidean