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A hyperbolic version of the Euclidean icosahedron.

A linear transformation A:R^n->R^n is hyperbolic if none of its eigenvalues has modulus 1. This means that R^n can be written as a direct sum of two A-invariant subspaces E^s ...

The metric ds^2=(dx^2+dy^2)/((1-x^2-y^2)^2) for the Poincaré hyperbolic disk, which is a model for hyperbolic geometry. The hyperbolic metric is invariant under conformal ...

A partial differential equation of second-order, i.e., one of the form Au_(xx)+2Bu_(xy)+Cu_(yy)+Du_x+Eu_y+F=0, (1) is called hyperbolic if the matrix Z=[A B; B C] (2) ...

A point p on a regular surface M in R^3 is said to be hyperbolic if the Gaussian curvature K(p)<0 or equivalently, the principal curvatures kappa_1 and kappa_2, have opposite ...

Also known as the a Lorentz transformation or Procrustian stretch, a hyperbolic transformation leaves each branch of the hyperbola x^'y^'=xy invariant and transforms circles ...

Taking the pole as the inversion center, the hyperbolic spiral inverts to Archimedes' spiral r=atheta.

The roulette of the pole of a hyperbolic spiral rolling on a straight line is a tractrix.

A substitution which can be used to transform integrals involving square roots into a more tractable form. form substitution sqrt(x^2+a^2) x=asinhu sqrt(x^2-a^2) x=acoshu

A 4-hyperboloid has negative curvature, with R^2=x^2+y^2+z^2-w^2 (1) 2x(dx)/(dw)+2y(dy)/(dw)+2z(dz)/(dw)-2w=0. (2) Since r=xx^^+yy^^+zz^^, (3) it follows that ...