As a title I have some trouble to understand the formula of curvature that professor gave to us during the lesson. I have already tried to read the book he suggested and searched on the internet but I did not find what I was looking for...

Firstable he gave us the definition of curvature using the arc lenght.

Given the curve $\gamma(t): [a,b] \to \mathbb{R}$, the curvature vector $\vec k(s)$ is defined as $\vec k(s)=\frac{d}{ds}\tau(s)=\frac{d^2}{ds^2}\gamma(s)$

Later, since usually arc lenght parametrization is less commonly used, he wrote down this:

Thank to the chain rule and the definition of function $s(t):[a,b] \to [0,L(\gamma)]$ in the arc lenght parametrization we obtain: $$\partial_s = \frac{1}{|\gamma'(t)|}\partial_t$$ Now we using the definition of curvature vector $\vec k(s)$: $$\partial_s^2 = \vec k = \frac{1}{|\gamma'(t)|}\partial_t \gamma_s = \frac{1}{|\gamma'(t)|} \left( \frac{\gamma'(t)}{|\gamma'(t)|} \right)' = \frac{\gamma''(t)- \langle \gamma''(t),\tau(t) \rangle \cdot \tau(t)}{|\gamma'(t)|^2} $$

I do not understand the last equality, esentially for two reasons.

  • The math process that the professor has followed to write down this result (I do not know how to do a derivate of a quotient in more than one variable)
  • I know that the derivate of a vector is another vector orthogonal to the first. For this reason I think that the scalar product should be zero.

In the end he said that $$P_{\gamma(x)}^{\perp}\left( \frac{\gamma''(t)}{|\gamma'(t)|^2} \right) = \frac{\gamma''(t)- \langle \gamma''(t),\tau(t) \rangle \cdot \tau(t)}{|\gamma'(t)|^2}$$ By definition: $$P_{\gamma(x)}^{\perp}(\eta) := \eta - \langle \eta, \tau \rangle \space \tau $$ where $\eta$ is a casual vector in $\mathbb{R^n}$ that has its application point on the curve; $\tau$ is the tangent vector at the point curve $x$. As the professor said $P_{\gamma(x)}^{\perp}(\eta)$ is the projection of vector $\eta$ to the normal vector of the curve point $x$, but I can not see why.

However I'm not pretty sure about the last definition and I can't find it nowhere (because I do not even know its name). Can somebody teach me what $P_{\gamma(x)}^{\perp}(\eta)$ is, please?


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