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This is a question from do Carmo exercises, Sec 2-5.

I know I just have to compute the area by using First Fundamental form, area $$ = \int\int\sqrt{EG-F^2}du dv$$
for a paramatrisation $x(u, v)$ of the tube surface, but I can't think of a parametrisation in just $2$ variables.

john doe
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1 Answers1

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I'm being too lazy to figure out which coordinate system your curve is in due to variable choice. If you have $\alpha: \mathbb{R} \to \mathbb{R}^3$ is regular, then at any point $\alpha(t)$ we know $\textbf{B}(t)$ and $\textbf{N}(t)$ form a unit-orthogonal basis at this point. Hence a tube of radius $r$ can be parametrized by,

$$x(u,v) = \alpha(u) +r\bigg( \textbf{N}(u)\cos(v) + \textbf{B}(u) \sin(v)\bigg)$$

Faraad Armwood
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