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 = [画像:-(sqrt(2)uvcosu)/(sqrt(1+u^2(2+v^2)+(1-u^2v^2)cos(2u)))]

(7)

f = [画像:-(sqrt(2)(cosu+usinu))/(sqrt(1+u^2(2+v^2)+(1-u^2v^2)cos(2u)))]

(8)

g = [画像:-(sqrt(2)u^2vsinu)/(sqrt(1+u^2(2+v^2)+(1-u^2v^2)cos(2u))),]

(9)

the area element is

[画像: 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 = [画像:(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 = [画像:-(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.html Pickover, C. A. Mazes for the Mind: Computers and the Unexpected. New York: St. Martin's Press, 1992.Wang, P. "Renderings." https://web.archive.org/web/19990913063231/http://www.ugcs.caltech.edu/~peterw/portfolio/renderings/.

Referenced on Wolfram|Alpha

Steinbach Screw

Cite this as:

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

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