Distinct distances in \(\mathbf{R}^3\) between quadratic and orthogonal curves (Q6568861)

The paper under review studies a variant of the Erdős distinct distance problem: assume \(n\) points are on one curve, \(m\) points are on another curve in the Euclidean plane. How many distinct distances must appear between the two point sets? The answer does depend on the curves.\N\NIf the curves are conic sections, a complete characterization is obtained for cases, where the number of distances is \(O(m+n)\). This includes new constructions for points on parabola vs. parabola, on ellipse vs. ellipse, and ellipse vs. hyperbola. In all other cases, the number of distances is \(\Omega(\min(m^{2/3}n^{2/3},m^2,n^2))\).\N\NIf the curves are not necessarily algebraic but smooth and are contained in perpendicular planes, a complete characterization is obtained for cases, where the number of distances is \(O(m + n)\). This includes a new construction of non-algebraic curves that involve logarithms. In all other cases, the number of distances is \(\Omega(\min(m^{2/3}n^{2/3},m^2,n^2))\).