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Elliptic Hyperboloid

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EllipticHyperboloid

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)/(a^2)+(y^2)/(b^2)-(z^2)/(c^2)=1,
(1)

and parametric equations

x(u,v)=asqrt(1+u^2)cosv
(2)
y(u,v)=bsqrt(1+u^2)sinv
(3)
z(u,v)=cu
(4)

for v in [0,2pi), or

x(u,v)=a(cosu∓vsinu)
(5)
y(u,v)=b(sinu+/-vcosu)
(6)
z(u,v)=+/-cv,
(7)

or

x(u,v)=acoshvcosu
(8)
y(u,v)=bcoshvsinu
(9)
z(u,v)=csinhv.
(10)

Taking the second of these with upper signs gives first fundamental form coefficients of

E=(a^2cos^2v+b^2sin^2v)/(4u)-1
(11)
F=1/4(b^2-a^2)sin(2v)
(12)
G=u(a^2sin^2v+b^2cos^2v),
(13)

second fundamental form coefficients of

e=(ab)/(2usqrt(a^2b^2+2u(a^2+b^2)+2(b^2-a^2)ucos(2v)))
(14)
f=0
(15)
g=(2abu)/(sqrt(a^2b^2+2u(a^2+b^2)+2(b^2-a^2)ucos(2v))).
(16)

The Gaussian curvature and mean curvature are

K=(4a^2b^2)/([a^2b^2+2u(a^2+b^2)+2(b^2-a^2)ucos(2v)]^2)
(17)
H=(ab(a^2+b^2+4u))/([a^2b^2+2u(a^2+b^2)+2(b^2-a^2)ucos(2v)]^(3/2)).
(18)

The Gaussian curvature can be giving implicitly by

K(x,y,z)=-(a^6b^6c^2)/((a^4b^4-a^2b^4x^2-b^4c^2x^2-a^4b^2y^2-a^4c^2y^2)^2)
(19)
=-(a^6b^2c^6)/((a^4c^4-a^2c^4x^2+b^2c^4x^2+a^4b^2z^2+a^4c^2z^2)^2)
(20)
=-(a^2b^6c^6)/((b^4c^4+a^2c^4y^2-b^2c^4y^2+a^2b^4z^2+b^4c^2z^2)^2).
(21)

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

(x^2)/(a^2)+(y^2)/(b^2)-(z^2)/(c^2)=-1,
(22)

and parametric equations

x=asinhucosv
(23)
y=bsinhusinv
(24)
z=c+/-coshu.
(25)

The two-sheeted elliptic hyperboloid oriented along the x-axis has Cartesian equation

(x^2)/(a^2)-(y^2)/(b^2)-(z^2)/(c^2)=1
(26)

and parametric equations

x=acoshucoshv
(27)
y=bsinhucoshv
(28)
z=csinhv.
(29)

The Gaussian curvature can be giving implicitly by

K(x,y,z)=(a^6b^6c^2)/((a^4b^4+a^2b^4x^2+b^4c^2x^2+a^4b^2y^2+a^4c^2y^2)^2)
(30)
=(a^6b^2c^6)/((a^4c^4+a^2c^4x^2-b^2c^4x^2-a^4b^2z^2-a^4c^2z^2)^2)
(31)
=(a^2b^6c^6)/((b^4c^4-a^2c^4y^2+b^2c^4y^2-a^2b^4z^2-b^4c^2z^2)^2).
(32)

See also

Hyperboloid, Ruled Surface

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References

Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 404-406 and 470, 1997.

Cite this as:

Weisstein, Eric W. "Elliptic Hyperboloid." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EllipticHyperboloid.html

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