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Sievert's Surface

SievertsSurface

A constant-curvature surface which can be given parametrically by

x=rcosphi
(1)
y=rsinphi
(2)
z=(ln[tan(1/2v)]+a(C+1)cosv)/(sqrt(C)),
(3)

where

phi=-u/(sqrt(C+1))+tan^(-1)(tanusqrt(C+1))
(4)
a=2/(C+1-Csin^2vcos^2u)
(5)
r=(asqrt((C+1)(1+Csin^2u))sinv)/(sqrt(C)),
(6)

with |u|<pi/2 and 0<v<pi (Reckziegel 1986).

The coefficients of the first fundamental form are

E=(64acos^2ucos^2v)/([4+3a-acos(2u)+2acos^2ucos^2(2v)]^2)
(7)
F=0
(8)
G=(64[(1+a)cscv+acos^2usinv]^2)/(4a[4+3a-acos(2u)+2acos^2ucos^2(2v)]^2),
(9)

and the coefficients of the second fundamental form are

e=sqrt(a/(a+1))(8acos^3usin(3v)-4cosu[8+11a+3acos(2u)])/([4+3a-acos(2u)+2acos^2ucos^2(2v)]^2)
(10)
f=0
(11)
g=sqrt((a+1)/a)([4+5a+acos(2u)-2acos^2ucos(2v)]csc(1/2v)sec(1/2v))/([4+3a-acos(2u)+2acos^2ucos^2(2v)]^2).
(12)

The Sievert surface has Gaussian and mean curvatures given by

K=1
(13)
H=1/(1+(a+1)tan^2u).
(14)

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References

Fischer, G. (Ed.). Plate 87 in Mathematische Modelle aus den Sammlungen von Universitäten und Museen, Bildband. Braunschweig, Germany: Vieweg, p. 83, 1986.Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 499-500, 1997.Reckziegel, H. "Sievert's Surface." §3.4.4.3 in Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, pp. 38-39, 1986.Sievert, H. Über die Zentralflächen der Enneperschen Flachen konstanten Krümmungsmaßes. Dissertation, Tübingen, 1886.

Cite this as:

Weisstein, Eric W. "Sievert's Surface." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SievertsSurface.html

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