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One has a hyperboloid of revolution if and only if =. Otherwise, the axes are uniquely defined (up to the exchange of the x-axis and the y-axis). There are two kinds of hyperboloids. In the first case (+1 in the right-hand side of the equation): a one-sheet hyperboloid, also called a hyperbolic hyperboloid.
For example, in thermodynamics the isothermal process explicitly follows the hyperbolic path and work can be interpreted as a hyperbolic angle change. Similarly, a given mass M of gas with changing volume will have variable density δ = M / V , and the ideal gas law may be written P = k T δ so that an isobaric process traces a hyperbola in the ...
The Weierstrass coordinates of a point are the Cartesian coordinates of the point when the point is mapped in the hyperboloid model of the hyperbolic plane, the x-axis is mapped to the (half) hyperbola ( , , +) and the origin is mapped to the point (0,0,1). [1]
Hyperbolic motions can also be described on the hyperboloid model of hyperbolic geometry. [ 1 ] This article exhibits these examples of the use of hyperbolic motions: the extension of the metric d ( a , b ) = | log ( b / a ) | {\displaystyle d(a,b)=\vert \log(b/a)\vert } to the half-plane and the unit disk .
One can take the hyperboloid to represent the events (positions in spacetime) that various inertially moving observers, starting from a common event, will reach in a fixed proper time. The hyperbolic distance between two points on the hyperboloid can then be identified with the relative rapidity between the two corresponding observers.
The Gudermannian function gives a direct relationship between the circular functions and the hyperbolic functions that does not involve complex numbers. The graph of the function a cosh(x/a) is the catenary, the curve formed by a uniform flexible chain, hanging freely between two fixed points under uniform gravity.
The isometry to the previous models can be realised by stereographic projection from the hyperboloid to the plane {+ =}, taking the vertex from which to project to be (, …,,) for the ball and a point at infinity in the cone () = inside projective space for the half-space.
Since the curvature is multiplied by we see that in this example the more (negatively) curved the space is, the lower the hyperbolicity constant. Similar examples are CAT spaces of negative curvature. While curvature is a property that is essentially local, hyperbolicity is a large-scale property which does not see local (i.e. happening in a ...