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Generating a Hyperboloid by Rotating a Line

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A hyperboloid can be generated by rotating a line around the axis. The vertical line through the origin is first translated in the - plane and then rotated about an axis in the - plane. Enough information is then available to use standard equations to calculate the foci and hyperbola curves; two foci are shown as small spheres.

Contributed by: Conrad A. Benulis (March 2011)
Open content licensed under CC BY-NC-SA

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The two vertices of the calculated hyperbola (green curves) are located on the transverse axis at . The point in the - plane is the initial translation point of the dynamic (red) line. The static angle in the - plane can be used to calculate the angle of the asymptotes with the transverse - axis, . The foci are then . The hyperbola equations are , , , where varies from 0° to 360°; is the conjugate axis . These calculations use only the initial dynamic rotating line position.

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