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hyperboloid    
双曲线体

双曲线体

hyperboloid
双曲线

hyperboloid
n 1: a quadric surface generated by rotating a hyperbola around
its main axis


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  • geometry - Hyperboloid Equation - Mathematics Stack Exchange
    This gives the equation of a hyperboloid produced by taking a hyperbola with foci F1 and F2 in some plane through the two foci and rotating that hyperbola about its transverse axis (the axis through the foci) The result is a hyperboloid of two sheets contained within a double cone The equation x2 a2 + y2 b2 − z2 c2 = 1,
  • What is the most accurate definition of the hyperboloid model of . . .
    Well the definition of the hyperbolic plane is not just a definition of a set Indeed the hyperboloid model is diffeomorphic to a Euclidean plane But not isometric! So, (one of) the (right) definition is: takes the component of the hyperboloid with x> 0 x> 0 and, as metric, restrict the Minkowski metric (that with signature ++-) (exercice: check that this is indeed positive definite) As for
  • Whats the metric on a 4-dimensional hyperboloid?
    Asking about "the" metric on a 4-dimensional hyperboloid, or even on the 2-dimensional one, does not make any sense There are a lot of metrics on hyperboloids
  • hyperbolic geometry - What is the metric of an hyperboloid . . .
    The sphere and the hyperboloid are connected like the sine and hyperbolic sine are connected Does that mean that the metric of a hyperboloid can be obtained by replacing the sine in the formula above with the hyperbolic sine?
  • differential geometry - parametrization of the hyperboloid of two . . .
    parametrization of the hyperboloid of two sheets Ask Question Asked 11 years, 3 months ago Modified 8 years, 2 months ago
  • Show that the hyperboloid of one sheet is a doubly ruled surface.
    11 Show that the hyperboloid of one sheet is a doubly ruled surface, i e each point on the surface is on two lines lying entirely on the surface (Hint: Write equation (1 35) as x2 a2 − z2 c2 = 1 − y2 b2 x 2 a 2 − z 2 c 2 = 1 − y 2 b 2, factor each side Recall that two planes intersect in a line
  • Shortest path on hyperboloid - Mathematics Stack Exchange
    On the sphere S2 S 2, the shortest path between two points is the great circle path How about H2 H 2, the hyperboloid x2 +y2 −z2 = −1, z ≥ 1 x 2 + y 2 − z 2 = − 1, z ≥ 1, with the Euclidean distance? Is there a formula for the shortest path between two points on the surface? And what is the length of the shortest path? Note that it is not the hyperbolic distance; it is the
  • calculus - Deriving parameterization for hyperboloid - Mathematics . . .
    I know there is a parameterization of a hyperboloid x2 a2 + y2 b2 − z2 c2 = 1 x 2 a 2 + y 2 b 2 − z 2 c 2 = 1 in terms of cosh cosh and sinh sinh, but I don't see how these equations are derived I would appreciate it if either someone could explain to me how such a parameterization is derived or recommend a reference
  • real analysis - Calculating the tangent space to a hyperboloid . . .
    Guillemin-Pollack 1 2 8: What is the tangent space to the hyperboloid defined by x2 +y2 −z2 = a at (a−−√, 0, 0), where a> 0? Is there a way to compute it using Characterization of the tangent space in terms of velocity vectors? I cannot apply the usual definition of Guillemin and Pollack because I don't know how exactly to parametrize a point with z = 0 on the hyperboloid (see





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