- #1

ozone

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## Homework Statement

Consider the tangent surface of some regular differentiable curve given as [itex]X(t,v) = \alpha(t) + v \alpha'(t) [/itex]. Show that the tangent planes along X(t,constant) are equal.

## Homework Equations

[itex] N = \frac{X_{t} \wedge X_{v}}{|X_{t} \wedge X_{v}|} [/itex]

The general tangent plane equation, [itex]T_{t_0}(S) = N|_{t_0, v_0} \cdot( (t - t_0) - (v-v_0)) = 0 [/itex]

## The Attempt at a Solution

Using the equation above I solved for [itex] N = \hat{v} b(s) [/itex] where b(s) is the binormal to the curve.

However I wasn't sure if this was correct since it does not seem like I am close to answering the problem. I wanted to reach the conclusion that [itex] \frac{dN}{dt} = 0 [/itex] along this curve, but this seemed to be untrue. Another idea was to show that the tangent plane equation is the same for all initial points [itex] t_0 [/itex] but this too seemed incorrect.

I am really stuck at this point and would love it if anyone could point me in the right direction