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What are some surfaces where $ \kappa_1-\kappa_2$ is constant? On a sphere where all are umbilical points.. is a special case.

For the $ \kappa_1+\kappa_2$ = constant case we have DeLaunay and Minimal surfaces.

$ \kappa_1,\kappa_2$ are the principal curvatures.

EDIT1:

In the limited number of numerical integrations done so far as a surface of revolution I obtain a symmetrical U shaped meridian between vertical asymptote planes in one case and two cuspidal ring edges in another.

Narasimham
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1 Answers1

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Another trivial case is a cylinder with $\kappa_{1}=\dfrac{1}{r}$ and $\kappa_{2}=0$.

How about another variation with $2\kappa_{1}-\kappa_{2}=0$?

Mylar balloon \begin{align*} \left( \begin{array}{c} x \\ y \\ z \\ \end{array} \right) &= r\left( \begin{array}{c} \text{cn} \left( u,\frac{1}{\sqrt{2}} \right) \cos v \\[5pt] \text{cn} \left( u,\frac{1}{\sqrt{2}} \right) \sin v \\[5pt] \sqrt{2} \left[ E \left( \text{am} \left( u,\frac{1}{\sqrt{2}} \right) ,\frac{1}{\sqrt{2}} \right) -\frac{1}{2} u \right] \end{array} \right) \\ \kappa_{1} &= \frac{1}{2r} \, \text{cn} \left( u,\frac{1}{\sqrt{2}} \right) \\ \kappa_{2} &= \frac{1}{r} \, \text{cn} \left( u,\frac{1}{\sqrt{2}} \right) \end{align*}

Ng Chung Tak
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  • Interesting. Curvature quotient rather than difference is constant here in the Elastica. The balloon has no circumferential stress, but only along its meridian – Narasimham Jul 25 '16 at 05:31
  • @Narasimham See also another article here. – Ng Chung Tak Aug 20 '19 at 10:38
  • Thanks for remembering and for the reference. DeLaunay Unduloids have been known;$\kappa_1/\kappa_2= 2 $ is Balloon case and $\kappa_1/\kappa_2= 1/2 $ is a common cycloid swept on x-axis.. I suppose unduloids have Gauss $K$ positive and negative in their parts. So $ a \kappa_1+ b \kappa_2= con $ is not another class at least for a surface of revolution. I was also looking for non-rotation symmetric surfaces.. – Narasimham Aug 21 '19 at 04:37