Circular arcs, circumference as arc length, and diameter

Let $c=(c_1,c_2)\in {\mathbb{R}}^2$ and $r>0$. A circular arc of the circle with centre $c$ and radius $r$ is a restriction

${\gamma }_{c,r}{\mathrm{\mid }}_{[\alpha ,\beta ]},{\gamma }_{c,r}(t)=c+r(\cos t,\sin t),\alpha <\beta ,$

taken as a parametrized path rather than as its image; its trace is that image. The parameter interval is part of the arc, because a set of points does not determine a length: the same trace is swept by restrictions of different lengths.

The arc's length is the path length of Paths in ${\mathbb{R}}^n$, inscribed polygonal sums, arc length as their supremum, and rectifiability computed with the Euclidean norm of The $p$-norms $\parallel x{\parallel }_p$ for rational $p\ge 1$, and $\parallel x{\parallel }_{\mathrm{\infty }}$. The circumference $C(r)$ is $L({\gamma }_{c,r})$ for the once-around parameter interval $[0,2\pi ]$, where $\pi$ is the constant of Pi as twice the smallest positive zero of cosine. Translation does not affect the value, so the notation suppresses $c$. The diameter is $d(r):=2r$.

The phrase once around is part of the convention: a parametrized path that repeats the same trace can have a larger length.