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Steinbach Screw

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SteinbachScrew

A surface generated by the parametric equations

x(u,v)=ucosv
(1)
y(u,v)=usinv
(2)
z(u,v)=vcosu.
(3)

The above image uses u in [-4,4] and v in [0,6.25].

The coefficients of the first fundamental form are

E=1+v^2sin^2u
(4)
F=-vcosusinu
(5)
G=1/2[1+2u^2+cos(2u)],
(6)

the coefficients of the second fundamental form are

e=-(sqrt(2)uvcosu)/(sqrt(1+u^2(2+v^2)+(1-u^2v^2)cos(2u)))
(7)
f=-(sqrt(2)(cosu+usinu))/(sqrt(1+u^2(2+v^2)+(1-u^2v^2)cos(2u)))
(8)
g=-(sqrt(2)u^2vsinu)/(sqrt(1+u^2(2+v^2)+(1-u^2v^2)cos(2u))),
(9)

the area element is

dA=sqrt((1+u^2(2+v^2)+(1-u^2v^2)cos(2u))/2)du ^ dv,
(10)

and the Gaussian and mean curvatures are given by

K=(4[u(u^2v^2-2)cosusinu-u^2sin^2u-cos^2u])/([1+u^2(2+v^2)+(1-u^2v^2)cos(2u)]^2)
(11)
H=-(v{u(5+4u^2)cosu-ucos(3u)})/(2sqrt(2)[1+u^2(2+v^2)+(1-u^2v^2)cos(2u)]^(3/2))-(2v[2+u^2(2+v^2)+(2-u^2v^2)cos(2u)]sinu)/(2sqrt(2)[1+u^2(2+v^2)+(1-u^2v^2)cos(2u)]^(3/2)).
(12)

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References

Update a linkNaylor, B. "Steinbach Screw 1." http://www.garlic.com/~bnaylor/raytrace/rtstein1.htmlPickover, C. A. Mazes for the Mind: Computers and the Unexpected. New York: St. Martin's Press, 1992.Update a linkWang, P. "Renderings." http://www.ugcs.caltech.edu/~peterw/portfolio/renderings/

Cite this as:

Weisstein, Eric W. "Steinbach Screw." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SteinbachScrew.html

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