# Can any curve in 3D space be described by an intersection of two surfaces?

Can any curve in 3D space be described by an intersection of two surfaces? If not, what assumptions I need to let it be true?

If this is too general, what if I restrict the scenarios to twice differentiable curves and surfaces? Suppose $\phi_i \in C^2 : \mathbb{R}^3 \rightarrow \mathbb{R}$. The two surfaces are represented by $\phi_1(x,y,z)=0$ and $\phi_2(x,y,z)=0$ respectively. Then the curve is the intersection of them: $L=\{(x,y,z) \;|\; \phi_1(x,y,z)=0, \phi_2(x,y,z)=0\}$.

• Of course it partly depends what is meant in terms of the smoothness of the curve and the smoothness of the surfaces. The question remains interesting if you restrict it to twice differentiable curves and surfaces. – Mark Fischler May 21 '18 at 20:02
• Yes, what would happen if I restrict to twice differentiable curves and surfaces? – winston May 21 '18 at 20:03
• I'm not sure if this is what you want, but, if $\gamma=(x(t),y(t),z(t))$ is a curve than the two surfaces $(x(t),y(t),z)$ and $(x,y(t),z(t))$ intersects on $\gamma$. – Emilio Novati May 21 '18 at 20:35
• This depends on what you mean by curve and surface. – Allawonder May 21 '18 at 21:55
• @Narasimham Is there a general method? – winston Jul 27 at 14:26

Not sure if I understood your question. Do you want to recover surfaces from their intersections ?

Before getting into DG, simpler cases should be understood.

If a cone and a plane intersect we have the conic sections. When two second order surfaces intersect the orthogonal projections of intersection are of second degree.

A helicoid and cylinder intersect to produce a helix. If the 3d helix is given, I cannot imagine what other surface pair would produce the same helix.

The space curve of intersection is of simple description only in simple cases.

For example if an ellipse of given axes size is given then the intersecting surface pair can be cone/plane, cylinder/plane etc.

If a circle is given some rotationally symmetric surface pairs are possible eg, Cylinder/sphere, Cone /ellipsoid etc. on same axis as shown in rough hand sketch would arise, the pair is not unique. 