  We characterize when two conic curved creases are compatible with each other,
when the rule lines must converge to conic foci and reflect at the crease.
Namely, two conics are compatible (can be connected by rule segments in a
foldable curved crease pattern) if and only if they have equal or reciprocal
eccentricity. Thus, circles (eccentricity 0) and parabolas (eccentricity 1) are
compatible with only themselves (when scaled from a focus), and ellipses
(eccentricity strictly between 0 and 1) and hyperbolas (eccentricity above 1)
are compatible with themselves and each other (but only in specific pairings).
The foundation of this result is a general condition relating any two curved
creases connected by rule segments. We also use our characterization to analyze
several curved crease designs.
