Hyperboloid

A Quadratic Surface which may be one- or two-sheeted.

The one-sheeted circular hyperboloid is a doubly Ruled Surface. When oriented along the z-Axis, the one-sheeted circular hyperboloid has Cartesian Coordinates equation

$${x^2\over a^2}+{y^2\over a^2}-{z^2\over c^2}=1,$$ (1)

and parametric equation

$$x = a\sqrt{1+u^2}\cos v$$ (2)

$$y = a\sqrt{1+u^2}\sin v$$ (3)

$$z = cu$$ (4)

for $v\in [0,2\pi)$ (left figure). Other parameterizations include

$$x(u,v) = a(\cos u\mp v\sin u)$$ (5)

$$y(u,v) = a(\sin u\pm v\cos u)$$ (6)

$$z(u,v) = \pm cv,$$ (7)

(middle figure), or

$$x(u,v) = a\cosh v\cos u$$ (8)

$$y(u,v) = a\cosh v\sin u$$ (9)

$$z(u,v) = c\sinh v$$ (10)

(right figure). An obvious generalization gives the one-sheeted Elliptic Hyperboloid.

A two-sheeted circular hyperboloid oriented along the z-Axis has Cartesian Coordinates equation

$${x^2\over a^2}+{y^2\over a^2}-{z^2\over c^2}=-1.$$ (11)

The parametric equations are

$$x = a\sinh u\cos v$$ (12)

$$y = a\sinh u\sin v$$ (13)

$$z = \pm c\cosh u$$ (14)

for $v\in [0,2\pi)$. Note that the plus and minus signs in $z$ correspond to the upper and lower sheets. The two-sheeted circular hyperboloid oriented along the x-Axis has Cartesian equation

$${x^2\over a^2}-{y^2\over a^2}-{z^2\over c^2}=1$$ (15)

and parametric equations

$$x = \pm a\cosh u\cosh v$$ (16)

$$y = a\sinh u\cosh v$$ (17)

$$z = c\sinh v$$ (18)

(Gray 1993, p. 313). Again, an obvious generalization gives the two-sheeted Elliptic Hyperboloid.

The Support Function of the hyperboloid of one sheet

$${x^2\over a^2}+{y^2\over b^2}-{z^2\over c^2}=1$$ (19)

is

$$h=\left({{x^2\over a^4}+{y^2\over b^4}+{z^2\over c^4}}\right)^{-1/2},$$ (20)

and the Gaussian Curvature is

$$K=-{h^4\over a^2b^2c^2}.$$ (21)

The Support Function of the hyperboloid of two sheets

$${x^2\over a^2}-{y^2\over b^2}-{z^2\over c^2}=1$$ (22)

is

$$h=\left({{x^2\over a^4}-{y^2\over b^4}+{z^2\over c^4}}\right)^{-1/2},$$ (23)

and the Gaussian Curvature is

$$K={h^4\over a^2b^2c^2}$$ (24)

(Gray 1993, pp. 296-297).

See also Catenoid, Ellipsoid, Elliptic Hyperboloid, Hyperboloid Embedding, Paraboloid, Ruled Surface

References

Fischer, G. (Ed.). Plates 67 and 69 in Mathematische Modelle/Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, pp. 62 and 64, 1986.

Gray, A. ``The Hyperboloid of Revolution.'' §18.5 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 296-297, 311-314, and 369-370, 1993.