We introduce a family of boundary conditions and point constraints for conformal immersions that increase the controllability of surfaces defined as minimizers of conformal variational problems. Our free boundary conditions fix the metric on the boundary, up to a global scale, and admit a discretization compatible with discrete conformal equivalence. We also introduce constraints on the conformal scale factor, enforcing rigidity of the geometry in regions of interest, and describe how in the presence of point constraints the conformal class encodes knot points of the spline that can be directly manipulated. To control the tangent planes, we introduce flux constraints balancing the internal material stresses. The collection of these point constraints provide intuitive controls for exploring a subspace of conformal immersions interpolating a fixed set of points in space. We demonstrate the applicability of our framework to geometric modeling, mathematical visualization, and form finding.

翻译：本文引入了一类用于共形浸入的边界条件与点约束，以增强由共形变分问题最小化解所定义曲面的可控性。我们的自由边界条件在全局尺度范围内固定边界上的度量，并允许一种与离散共形等价相容的离散化方案。我们还引入了对共形尺度因子的约束，以强制感兴趣区域的几何刚性，并描述了在存在点约束时共形类如何编码可直接操作的样条节点。为控制切平面，我们引入了平衡内部材料应力的通量约束。这些点约束的集合为探索空间中固定点集插值的共形浸入子空间提供了直观的控制手段。我们展示了该框架在几何建模、数学可视化和形态寻找中的适用性。