The Bohemian Dome
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Introduction:
One could imagine creating a torus by taking a small ellipse and rotating it around a larger circle, smoothly turning the small ellipse to form a nice even tube. If, instead of turning the small ellipse as we rotate it, we force it to remain parallel to a certain plane (the yz plane in the plot above), the surface obtained is called a bohemian dome. The bohemian dome is one example of a type of surface known as a translation surface, the surface that results when one curve is rotated about a second curve in such a way that some point on the first curve traces the second curve.
The structure of the bohemian dome is easier to see when it is presented in a cutaway view:
Definition:
In this case, a is the radius of the larger circle, and b and c are the lengths of the axes of the ellipse.
Properties:
Tangent Planes:At u = u0, v = v0, the tangent plane to the surface is parameterized by:
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Infinitesimal Area:The infinitesimal area of a patch on the surface is given by |
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Gaussian Curvature:
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Mean Curvature:
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