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Joining Two Pairs of Planar Points by Disjoint Arcs: The Sharp Length Bound
George M. Georgiou
Problem F16 of Croft, Falconer, and Guy asks for the least worst-case length needed to join prescribed pairs of points by pairwise disjoint planar arcs, when the distance within ea…
Five-Point Hyperbolas on Power Curves and Near Lamé-Oval Vertices
George M. Georgiou
Problem A33 of Croft--Falconer--Guy records Reznick's question about infinite plane sets for which every five-point subset determines an ellipse or, respectively, a hyperbola, and…
Reciprocal-Polar Linearization and Two Conic Constructions for the Cone Projection
George M. Georgiou
Let $$ f_R(x)=\frac{x}{1+\norm{x}/R},\qquad R>0, $$ be the radial cone projection of the plane onto the open disk $\DR=\{x:\norm{x}<R\}$. Previous work established the Self-Directr…
Straightedge-and-Compass Constructibility of the Reciprocal--th-Power Law
George M. Georgiou
Given positive lengths and and a positive integer , let be determined by We prove that a single finite unmarked-stra…