Question

Theorem
Let α(s) be parametrized by arc length. Then |α′(s)| =1. (Prove it)

Answer 1

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Answer on Question #70540, Math / Geometry

Theorem. Let $\alpha(s)$ be parametrized by arc length. Then $|\alpha'(s)| = 1$.

Proof. Let $\varphi: I \to \mathbf{P}^3$ is a regular curve and the arc length is $s(t) = \int_{t_0}^{t} |\varphi'(t)| dt$. We solve for $t$ as $t = t(s)$ to get the function $t: J \to I$. Then

$$\alpha(s) = \varphi(t(s))$$

is parametrized by arc length. So,

$$|\alpha'(s)| = \left| \frac{d\alpha}{ds} \right| = \left| \frac{d\alpha}{dt} \cdot \frac{dt}{ds} \right| = \left| \frac{d\varphi(t(s))}{dt} \cdot \frac{1}{\frac{ds}{dt}} \right| = \left| \frac{d\varphi(t)}{dt} \cdot \frac{1}{\varphi'(t)} \right| = 1.$$