I need to verify the Serret-Frenet equations for $ \gamma(t) = (4/5 \cos t, 1-\ \sin t, -3/5 \cos t)$ That is I need to verify $\dot t = \kappa n, \dot n = -\kappa t+ \tau b, \dot b = -\tau n$ Here from the given $\gamma(t)$ I can find $t$ and $\dot t$. But how to find $n$? If I use the relation $n=\dot t/{\kappa}$ won’t it be a circular argument since that is what I need to verify?
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Both $t, n, b$ are defined independent of the $3$ equations you have. – Arctic Char Sep 22 '20 at 10:27
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For clarity, we need to use different notation for $t$ vector and $t$ as a parameter.
By definition we have that
$$T=\frac{\dot{\gamma(t)}}{\left|\dot{\gamma(t)}\right|}$$
and
$$N=\frac{\dot{T}}{\left|\dot{T}\right|}=\frac{\ddot{\gamma(t)}}{\left|\ddot{\gamma(t)}\right|}$$
with $N\cdot T=0$, indeed
$$\frac d{dt}(T\cdot T)=2T\cdot \dot T=0$$
Refer also to the related
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Where is this definition from? I’m seeing principal normal defined in terms of curvature and also curvature is defined as the norm of derivative of tangent and I’m again back to the circular argument. – danny Sep 22 '20 at 10:45
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Thanks but these are unit principal normal and unit tangent vector right? How can I simply find principal normal without it being a unit principal normal? Will it be just the derivative of the tangent vector? – danny Sep 22 '20 at 11:03
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