Astroidal Ellipsoid

AstroidalEllipsoid

The surface which is the inverse of the ellipsoid in the sense that it "goes in" where the ellipsoid "goes out." It is given by the parametric equations

x = acos^3ucos^3v

(1)

y = bsin^3ucos^3v

(2)

z = csin^3v

(3)

for u in [-pi/2,pi/2] and v in [-pi,pi].

The special case a=b=c corresponds to the hyperbolic octahedron. Like the hyperbolic octahedron, the astroidal ellipse is an algebraic surface of degree 18 with very complicated terms.

The astroidal ellipsoid has first fundamental form coefficients

E = 9cos^2ucos^6vsin^2u(a^2cos^2u+b^2sin^2u)

(4)

F = 9cosucos^5vsinu(a^2cos^4u-b^2sin^4u)sinv

(5)

G = 9(a^2cos^6u+b^2sin^6u)sin^2vcos^4v+9c^2sin^4vcos^2v,

(6)

while the coefficients of the second fundamental form are more complicated.

K=(a^2b^2c^2sec^4v)/(9[a^2(b^2cos^2vsin^2u+c^2sin^2v)cos^2u+b^2c^2sin^2usin^2v]^2),

(7)

while the mean curvature has a complicated expression.

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More things to try:

Nordstrand, T. "Astroidal Ellipsoid." http://jalape.no/math/asttxt.htm.