Quadrilaterals inscribed in a Jordan curve Missing points, arc balancing, and collapses Rectangles, rhombi, and families of parallelograms

We study the points of a planar Jordan curve that are not vertices of any inscribed quadrilateral in a given class. A compact space of quadruples dividing the arcs into two equal masses carries a rank-three section whose degree modulo two is one. The same construction applies to rectangles and rhombi. Concentrating five masses recovers Schwartz’s bound of four points missing from rectangles, with a lower bound on both side lengths; three masses give at most two points missing from graceful rhombi. This latter bound is attained on an explicit curve. Missing points impose distributions of vertices among arcs and continua of configurations. We establish a continuation between two limiting chords in the case of four points missing from rectangles, and between two collapses in the case of two points missing from rhombi. For every finite real parameter λ, the metric family of parallelograms Pλ likewise has at most two missing points. Finally, homotopies toward squares localize losses of compactness, and a counterexample distinguishes quadrilateral absence from triangular absence. The arguments apply in the C0 setting; finite counting statements are distinguished from degree arguments. The same degree directly recovers Schwartz’s result: three points missing from rectangles force all aspect ratios.