  Motivated by the prospect of measuring a black hole photon ring, in previous
work we explored the interferometric signature produced by a bright, narrow
curve in the sky. Interferometric observations of such a curve measure its
"projected position function" $\mathbf{r}\cdot\hat{\mathbf{n}}$, where
$\mathbf{r}$ parameterizes the curve and $\hat{\mathbf{n}}$ denotes its unit
normal vector. In this paper, we show by explicit construction that a curve can
be fully reconstructed from its projected position, completing the argument
that space interferometry can in principle determine the detailed photon ring
shape. In practice, near-term observations may be limited to the visibility
amplitude alone, which contains incomplete shape information: for convex
curves, the amplitude only encodes the set of projected diameters (or "widths")
of the shape. We explore the freedom in reconstructing a convex curve from its
widths, giving insight into the shape information probed by technically
plausible future astronomical measurements. Finally, we consider the Kerr
"critical curve" in this framework and present some new results on its shape.
We analytically show that the critical curve is an ellipse at small spin or
inclination, while at extremal spin it becomes the convex hull of a Cartesian
oval. We find a simple oval shape, the "phoval", which reproduces the critical
curve with high fidelity over the whole parameter range.
