# Thread: geodesic is parametrised by a constant multiple of arc length

1. ## geodesic is parametrised by a constant multiple of arc length

Let $S\in \mathbb{R}^3$ be a regular surface, and $\gamma: I \rightarrow S$ a geodesic, where $I$ is an open interval in $\mathbb{R}$. We have that $\frac{d}{dt}|\gamma ' (t) | = 0 \implies |\gamma '(t)| = c$, a constant , i.e.

$
\frac{ds}{dt} = | \gamma' (t) | = c$
..................(1)

where s is the arc length parameter $s(x) = \int_a^x |\gamma'(p)| dp$, $a\in I$. I understand what is happening up to here but then my notes claim that $t=\frac{s}{c}$. However do we not get that upon rearranging (1)

$ds=cdt \implies s = ct + A$ where A is some constant? Why is A zero? Any help would be appreciated

2. The constant is omitted in your notes, though it shouldn't be. Affine transformations of the parameter do not alter geodesy.

3. The choice of the added constant, A, is just a choice of where to start measuring the arclength. It is always possible to choose the starting point so that A= 0. I expect that it was left out here because the starting point is not relevant.

4. it was left out here because the starting point is not relevant