Associate family
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 x k ( ζ ) = ℜ { ∫ 0 ζ φ k ( z ) d z } + c k , k = 1 , 2 , 3 {\displaystyle x_{k}(\zeta )=\Re \left\{\int _{0}^{\zeta }\varphi _{k}(z)\,dz\right\}+c_{k},\qquad k=1,2,3} the family is described by x k ( ζ , θ ) = ℜ { e i θ ∫ 0 ζ φ k ( z ) d z } + c k , θ ∈ [ 0 , 2 π ] {\displaystyle x_{k}(\zeta ,\theta )=\Re \left\{e^{i\theta }\int _{0}^{\zeta }\varphi _{k}(z)\,dz\right\}+c_{k},\qquad \theta \in [0,2\pi ]} where ℜ {\displaystyle \Re } indicates the real part of a complex number.