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Hyperbola: the midpoints of parallel chords lie on a line. Hyperbola: the midpoint of a chord is the midpoint of the corresponding chord of the asymptotes. The midpoints of parallel chords of a hyperbola lie on a line through the center (see diagram). The points of any chord may lie on different branches of the hyperbola.
The discussion of plane sections can be performed for the unit hyperboloid of two sheets with equation : + = which can be generated by a rotating hyperbola around one of its axes (the one that cuts the hyperbola) A plane with slope less than 1 (1 is the slope of the asymptotes of the generating hyperbola) intersects either in an ellipse or in a ...
Hyperboloid of one sheet, such as cooling towers A hyperboloid of one sheet is a doubly ruled surface , and it may be generated by either of two families of straight lines. The hyperbolic paraboloid is a doubly ruled surface so it may be used to construct a saddle roof from straight beams.
The vectors v ∈ R n+1 such that Q(v) = -1 form an n-dimensional hyperboloid S consisting of two connected components, or sheets: the forward, or future, sheet S +, where x 0 >0 and the backward, or past, sheet S −, where x 0 <0. The points of the n-dimensional hyperboloid model are the points on the forward sheet S +.
A hyperbolic sector is a region of the Cartesian plane bounded by a hyperbola and two rays from the origin to it. For example, the two points (a, 1/a) and (b, 1/b) on the rectangular hyperbola xy = 1, or the corresponding region when this hyperbola is re-scaled and its orientation is altered by a rotation leaving the center at the origin, as with the unit hyperbola.
There are an infinite number of uniform tilings based on the Schwarz triangles (p q r) where 1/p + 1/q + 1/r < 1, where p, q, r are each orders of reflection symmetry at three points of the fundamental domain triangle, the symmetry group is a hyperbolic triangle group. There are also infinitely many uniform tilings that cannot be generated from ...
Most hyperbolic surfaces have a non-trivial fundamental group π 1 = Γ; the groups that arise this way are known as Fuchsian groups. The quotient space H 2 / Γ of the upper half-plane modulo the fundamental group is known as the Fuchsian model of the hyperbolic surface. The Poincaré half plane is also hyperbolic, but is simply ...
If P 0 is taken to be the point (1, 1), P 1 the point (x 1, 1/x 1), and P 2 the point (x 2, 1/x 2), then the parallel condition requires that Q be the point (x 1 x 2, 1/x 1 1/x 2). It thus makes sense to define the hyperbolic angle from P 0 to an arbitrary point on the curve as a logarithmic function of the point's value of x .