# About the definition of curvature

In Do Carmo's differential geometry book, he says for a curve $$\alpha: I=(a,b)\rightarrow\mathbb{R}^3$$ parametrized by arc length, "since the tangent vector $$\alpha'$$(s) has unit length, the norm $$|\alpha''(s)|$$ of the second derivative measures the rate of change of the angle which neighboring tangents make with the tangent at $$s$$.

Why does the unit length of the tangent vector imply this geometric meaning of $$|\alpha''(s)|$$?

In geometry, we're interested in this second type of change. We don't want the change in the magnitude of velocity counts because we want a straight line to have zero curvature. Therefore, we must first do something to ensure that the velocity of our curve is always constant, preferably equal to $$1$$. This can be achieved by reparametrizing our curve using the arc length as you said. See here for more information about reparametrizing by the arc length.
It is easier to think of it in two dimensions. Suppose $$\alpha: I \rightarrow \mathbb{R}^2$$. We can encode the derivative with polar coordinates. There are two functions $$r:I\rightarrow\mathbb{R}$$ and $$\theta:I\rightarrow\mathbb{R}$$ such that $$\alpha'(s) = (r(s)\cdot \cos \theta(s), r(s)\cdot \sin \theta(s)).$$ Notice that \begin{align} \alpha''(s) &= (r'(s)\cdot \cos \theta(s) - r(s)\theta'(s) \sin \theta(s), r'(s)\cdot \sin \theta(s) + r(s)\theta'(s) \cos \theta(s)) \\ &= r'(s) (\cos \theta(s), \sin \theta(s)) + r(s)\theta'(s) (-\sin \theta(s), \cos \theta(s)). \end{align} The first term is the forward acceleration and the second term is the centripetal acceleration. If we only want the rate that the angle is changing, $$\theta'(s)$$, then we could force $$r(s)$$ to be 1 by reparametrizing the curve. If we set $$r(s)=1$$, then \begin{align} \alpha''(s) &= r'(s) (\cos \theta(s), \sin \theta(s)) + r(s)\theta'(s) (-\sin \theta(s), \cos \theta(s)) \\ &= \theta'(s) (-\sin \theta(s), \cos \theta(s)). \end{align} Taking the norm of both sides gives, \begin{align} ||\alpha''(s)||&= ||\theta'(s) (-\sin \theta(s), \cos \theta(s)) || \\ &= |\theta'(s)|\cdot ||(-\sin \theta(s), \cos \theta(s)) || \\ &= |\theta'(s)|. \end{align}