I am having trouble understanding some steps in this procedure:
Arc length $l(p)$ of ${\vec{X}(q)|q \in [a,p]}$ is: $$l(p) = \int_{a}^{p} \sqrt{X'(p)^2 +Y'(p)^2}dq$$
Given s in $[0,1]$, find p such that $\vec{X(p)} = \vec{x(s)}$ amounts to solving the nonlinear scalar equation $$f(p) = l(p) - s l(b) = 0 (1)$$
by using Newton's method.
Where does equation (1) come from and why do we have to find such a $q$
$\newcommand{\Reals}{\mathbf{R}}$Let $X:[a, b] \to \Reals^{n}$ be a piecewise-$C^{1}$ path, $$ \ell(b) = \int_{a}^{b} \|X'(q)\|\, dq $$ the arc length of $X$. The goal is reparametrize $X$ using $s:[0, 1] \to [a, b]$ so that $x(t) = (X \circ s)(t)$ has constant speed $\|x'(t)\| = \ell(b)$. In other words, given a real number $s$ with $0 \leq s \leq 1$, the goal is to find the real number $p$ with $a \leq p \leq b$ such that $$ \ell(b) s = \int_{0}^{s} \|x'(t)\|\, dt = \int_{a}^{p} \|X'(q)\|\, dq = \ell(p). $$ Rearranging, $f(p) := \ell(p) - \ell(b) s = 0$.