Abstract
We study the minimum number of distinct distances between point sets on two curves in R3. 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 Ω(min{m2/3n2/3,m2,n2}).(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 Ω(min{m2/3n2/3,m2,n2}).
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