The projection of the cubic 3D curve on each plane is still a cubic curve, so just solve all three equations and interpolate: (a1x^3+b1x^2+c1x+d1=0) (a2y^3+b2y^2+c2y+d2=0) (a3z^3+b3z^2+c3z+d3=0) But there are probably hundreds of interpolation algorithms.