Distinct distances in R3 between quadratic and orthogonal curves

Abstract

We study the minimum number of distinct distances between point sets on two curves in $R^3$. Assume that one curve contains $m$ points and the other $n$ points. Our main results:

(a) When the curves are conic sections, we characterize all cases where the number of distances is $O(m+n)$. This includes new constructions for points on two parabolas, two ellipses, and one ellipse and one hyperbola. In all other cases, the number of distances is $\Omega (min\{m^{2/3}{n}^{2/3},m^2,n^2\})$.

(b) When the curves are not necessarily algebraic but smooth and contained in perpendicular planes, we characterize all cases where the number of distances is $O(m+n)$. This includes a surprising new construction of non-algebraic curves that involve logarithms. In all other cases, the number of distances is $\Omega (min\{m^{2/3}{n}^{2/3},m^2,n^2\})$.