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This property is called Wren's theorem. [1] The more common generation of a one-sheet hyperboloid of revolution is rotating a hyperbola around its semi-minor axis (see picture; rotating the hyperbola around its other axis gives a two-sheet hyperbola of revolution). A hyperboloid of one sheet is projectively equivalent to a hyperbolic paraboloid.
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 quadrant of absolute temperature and gas density. For hyperbolic coordinates in the theory of relativity see the History section.
A hyperbola and its conjugate may have diameters which are conjugate. In the theory of special relativity, such diameters may represent axes of time and space, where one hyperbola represents events at a given spatial distance from the center, and the other represents events at a corresponding temporal distance from the center.
Subsets of the theory of hyperbolic groups can be used to give more examples of hyperbolic spaces, for instance the Cayley graph of a small cancellation group. It is also known that the Cayley graphs of certain models of random groups (which is in effect a randomly-generated infinite regular graph) tend to be hyperbolic very often.
The orthogonal group O(1, n) acts by norm-preserving transformations on Minkowski space R 1,n, and it acts transitively on the two-sheet hyperboloid of norm 1 vectors. Timelike lines (i.e., those with positive-norm tangents) through the origin pass through antipodal points in the hyperboloid, so the space of such lines yields a model of ...
The case a = 1 is called the unit hyperbola. The conjugate hyperbola is given by {: ‖ ‖ =} with an upper and lower branch passing through (0, a) and (0, −a). The hyperbola and conjugate hyperbola are separated by two diagonal asymptotes which form the set of null elements:
The Three Dunce Caps Theorem then says that P 1, P 2, and P 3 all lie on the same line. [4] Proof: Construct a sphere on top of each circle and then construct a plane tangent to these three spheres. The plane intersects the plane that the circles lies on at a straight line containing P 1, P 2, and P 3. These points are also the centers of ...
A ray through the unit hyperbola x 2 − y 2 = 1 at the point (cosh a, sinh a), where a is twice the area between the ray, the hyperbola, and the x-axis. For points on the hyperbola below the x-axis, the area is considered negative (see animated version with comparison with the trigonometric (circular) functions).