# Design of tangent vector fields

ACM Trans. Graphics (SIGGRAPH), 26(3), 2007.

Interactive control of direction fields for real-time surface texture synthesis.

**Abstract:**
Tangent vector fields are an essential ingredient in controlling surface appearance for applications
ranging from anisotropic shading to texture synthesis and non-photorealistic rendering. To achieve a
desired effect one is typically interested in smoothly varying fields that satisfy a sparse set of
user-provided constraints. Using tools from Discrete Exterior Calculus, we present a simple and efficient
algorithm for designing such fields over arbitrary triangle meshes. By representing the field as scalars
over mesh edges (i.e. discrete 1-forms), we obtain an intrinsic, coordinate-free formulation in which
field smoothness is enforced through discrete Laplace operators. Unlike previous methods, such a
formulation leads to a linear system whose sparsity permits efficient pre-factorization. Constraints are
incorporated through weighted least squares and can be updated rapidly enough to enable interactive design,
as we demonstrate in the context of anisotropic texture synthesis.

**Hindsights:**
One generalization is `N`-symmetry fields as in
Palacios and Zhang 2007 and
Ray et al 2008.

A common drawback in most linear approaches (including ours) is that the objective functional
under-penalizes smoothness of the vector field in regions where the vectors have small magnitude.
The work of
Crane et al. 2010
overcomes this by optimizing over direction angles instead of vectors.