Let $r$ be a natural number, and consider the $\mathbb{Z}^2$ lattice points $S$ inside or on the circle $C$ of radius $r$ centered on the origin. Let $P$ be the convex hull of $S$; so $P$ is inscribed in $C$. I would like to "triangulate" $P$ in a special sense.
A triangulation of $P$ is a partition of $P$ into triangles whose interiors are pairwise disjoint, whose corners are points of $S$, and such that every point of $S$ is on the boundary of a triangle—either a corner or on the interior of an edge.
Define the weight of a triangulation of $P$ as the sum of the Euclidean lengths of the segments comprising the triangulation. (Each segment is counted once even if shared between two triangles.)
Q. What is the minimum weight triangulation of $P$ as a function of $r$?
For $r=1$, the shortest (minimum weight) triangulation has length $2 + 4 \sqrt{2} \approx 7.66$—a split diamond. But even for $r=2$, the minimum length is not obvious (to me). Here are four different triangulations for $r=2$ (where $S$ is a subset of a $5 \times 5$ grid) and their associated lengths. (Pardon any calculation errors.)
$r=2$: $8+12 \sqrt{2} \approx 24.97$.
Below is just one triangulation for $r=3$, where $S$ is a subset of a $7 \times 7$ grid:
$r=3$: $26+4 \sqrt{2}+8 \sqrt{5}+4 \sqrt{10} \approx 62.19$.
I am not seeing an obvious pattern. Has this been investigated in the literature? Are there at least asymptotic bounds?
Update (10Sep2018). Here are @WlodekKuperberg's shorter triangulations for $r=2,3$:
Update (11Sep2018). Here is one triangulation for $r=4$, in a $9 \times 9$ grid:
$r=4$: $44+12 \sqrt{2}+8 \sqrt{5}+4 \sqrt{17} \approx 95.35$.