In differential geometry, the associate family (or Bonnet family) of a minimal surface is a one-parameter family of minimal surfaces which share the same Weierstrass data. That is, if the surface has the representation
xk(\zeta)=\Re\left\{
| \zeta | |
| \int | |
| 0 |
\varphik(z)dz\right\}+ck, k=1,2,3
xk(\zeta,\theta)=\Re\left\{ei
| \zeta | |
| \int | |
| 0 |
\varphik(z)dz\right\}+ck, \theta\in[0,2\pi]
where
\Re
For θ = π/2 the surface is called the conjugate of the θ = 0 surface.[1]
The transformation can be viewed as locally rotating the principal curvature directions. The surface normals of a point with a fixed ζ remains unchanged as θ changes; the point itself moves along an ellipse.
Some examples of associate surface families are: the catenoid and helicoid family, the Schwarz P, Schwarz D and gyroid family, and the Scherk's first and second surface family. The Enneper surface is conjugate to itself: it is left invariant as θ changes.
Conjugate surfaces have the property that any straight line on a surface maps to a planar geodesic on its conjugate surface and vice versa. If a patch of one surface is bounded by a straight line, then the conjugate patch is bounded by a planar symmetry line. This is useful for constructing minimal surfaces by going to the conjugate space: being bound by planes is equivalent to being bound by a polygon.[2]
There are counterparts to the associate families of minimal surfaces in higher-dimensional spaces and manifolds.[3]