Extrinsically smooth direction fields

We consider the problem of finding a unit vector field (i.e.; a direction field) over a domain that balances two competing objectives, smoothness and conformity to the shape of the domain. Common examples of this problem are finding normal directions along a curve and tangent directions over a surface. In a recent work, Jakob et al. observed that minimizing extrinsic variation of a tangent direction field on a surface achieves both objectives without the need for parameter-tuning or the use of additional constraints. Inspired by their empirical observations, we analyze the relation between extrinsic smoothness, intrinsic smoothness, and shape conformity in a continuous and general setting. Our analysis not only explains their observations but also suggests that an extrinsically smooth normal field along a curve can strike a similar balance between smoothness and shape-awareness. Our second contribution is offering extension of, justification for and improvement over the optimization framework of Jakob et al. In our experiments, we demonstrate the suitability of extrinsically smooth field in a variety of applications and compared with existing solutions.