Elliptic Hyperboloid

The elliptic hyperboloid is the generalization of the Hyperboloid to three distinct semimajor axes. The elliptic hyperboloid of one sheet is a Ruled Surface and has Cartesian equation

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

and parametric equations

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

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

$$z(u,v) = cu$$ (4)

for $v \in [0, 2\pi)$, or

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

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

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

or

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

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

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

The two-sheeted elliptic hyperboloid oriented along the z-Axis has Cartesian equation

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

and parametric equations

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

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

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

The two-sheeted elliptic 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

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

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

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

See also Hyperboloid, Ruled Surface

References

Gray, A. Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 296-297, 1993.