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So a developable surface can be parametrized as

$x(s, t)=\alpha(s)+t \beta(s)$

I can see that $\beta(s)$ is the direction of the principal curvature plane with k=0, but why is it the minimum or maximum curvature plane cutting through that point? Is $\alpha(s)$ a plane curve on the other principal curvature plane?

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  • $\begingroup$ What is definition of developable surface that you use? Wikipedia defines it precisely as surface with zero curvature $\endgroup$ – Blazej Sep 26 '15 at 8:18
  • $\begingroup$ I'm using this definition: x(s,t)=α(s)+tβ(s) $\endgroup$ – UXkQEZ7 Sep 26 '15 at 8:36
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The vector you have given $x(s, t)=\alpha(s)+t \beta(s)$ is a ruled surface with generator $ \beta(s)$.

It may or may not be developable depending on tangent vector triple product.

It is developable if $ (T, \beta(s),\beta{'}(s)) = 0 $

and skew ( twisted with negative Gauss curvature K) if

$(T, \beta(s),\beta{'}(s)) \ne 0. $

$ K = k_1\cdot k_2 = 0 $ is necessary and sufficient condition. When parametric lines of principal curvature $k_1=0 $ for $K=0$ then that parameter defines the straight edge or regression line of a developable surface.

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