Distinct distances in R3 between quadratic and orthogonal curves
European Journal of Combinatorics(2024)
摘要
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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