Schwarz singularities at focal points for analytic curves

Let $\Gamma$ be an analytic curve, and let $R$ denote the radius of curvature at a point of highest curvature on $\Gamma$. A Schwarz singularity is a singularity of the analytic continuation associated with the Schwarz function of the curve; in the parabola example it is of square-root type, and the concave side is the side toward which the curve bends. Schwarz singularities at focal points. For a general analytic curve, a square-root type Schwarz singularity appears near the point of highest curvature, approximately at distance $R/2$ on the concave side; this approximation becomes exact as $R\to 0$. This conjecture is motivated by the exact focal-point behavior for parabolas and by the fact that the foci of ellipses induce singularities, and is supported numerically for other analytic curves. The claim is stated only in an approximate sense away from the limit $R\to 0$.