Exploring Local Modifications for Constrained Meshes

Computer Graphics Forum 32(2) (Proc. Eurographics), 2013


Mesh editing under constraints is a challenging task with numerous applications in geometric modeling, industrial design, and architectural form finding. Recent methods support constraint-based exploration of meshes with fixed connectivity, but commonly lack local control. Because constraints are often globally coupled, a local modification by the user can have global effects on the surface, making iterative design exploration and refinement difficult. Simply fixing a local region of interest a priori is problematic, as it is not clear in advance which parts of the mesh need to be modified to obtain an aesthetically pleasing solution that satisfies all constraints.
We propose a novel framework for exploring local modifications of constrained meshes. Our solution consists of three steps. First, a user specifies target positions for one or more vertices. Our algorithm computes a sparse set of displacement vectors that satisfies the constraints and yields a smooth deformation. Then we build a linear subspace to allow realtime exploration of local variations that satisfy the constraints approximately. Finally, after interactive exploration, the result is optimized to fully satisfy the set of constraints. We evaluate our framework on meshes where each face is constrained to be planar.


Handle-based local modifications of a PQ mesh, showing the tradeoff between fairness and sparsity. Left: the input displacement for the original mesh. Right: results with increasing fairness weight from left to right. The color-coding shows the lengths of vertex displacements, clamped and normalized by the input displacement length for the handle vertex.

Local modifications of a planar tri-hex model (left) and a planar hex model (right). The input displacements are shown on the top.

Subspace exploration of local modifications for a PQ mesh. Left: input displacements with the corresponding handle-based local deformations. Right: local modifications using different linear combinations of the subspace basis vectors.<