  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
$\Omega(\min\{m^{2/3}n^{2/3},m^2,n^2\})$.
