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Rotational Symmetry Field Design on Surfaces

Jonathan Palacios and Eugene Zhang
ACM Transactions on Graphics, Vol 26(3), (SIGGRAPH 2007), 2007, Aritcle No. 55.

Paper (PDF, 3.75 Mb).

Video (MOV, 92.6 Mb).

Talk slides (PPT, 22.8 Mb).

This material is based upon work supported by the National Science Foundation under Grant No. CCF-0546881.

Any opinions, findings and conclusions or recomendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation (NSF).


Designing rotational symmetries on surfaces is a necessary task for a wide variety of graphics applications, such as surface
parameterization and remeshing, painterly rendering and pen-and-ink sketching, and texture synthesis. In these applications, the topology of a rotational symmetry field such as singularities and separatrices can have a direct impact on the quality of the results. In this paper, we present a design system that provides control over the topology of rotational symmetry fields on surfaces.

As the foundation of our system, we provide comprehensive analysis for rotational symmetry fields on surfaces and present efficient algorithms to identify singularities and separatrices. We also describe design operations that allow a rotational symmetry field to be created and modified in an intuitive fashion by using the idea of basis fields and relaxation. In particular, we provide control over the topology of a rotational symmetry field by allowing the user to remove singularities from the field or to move them to more desirable locations.

At the core of our analysis and design implementations is the observation that $N$-way rotational symmetries can be described by symmetric $N$-th order tensors, which allows an efficient vector-based representation that not only supports coherent definitions of arithmetic operations on rotational symmetries but also enables many analysis and design operations for vector fields to be adapted to rotational symmetry fields.

To demonstrate the effectiveness of our approach, we apply our design system to pen-and-ink sketching and geometry remeshing.


1. Natural rotational symmetries found in the five Platonic solids.


2. Topological editions opeartions for pen-and-ink sketching of the Bimba. From left to middle, a singularity pair is cancelled near the neckline. From middle to right, a singularity (in the highlight region near the corner of the right eye) is moved within the region.


3. A 6-RoSy field on a dragon highlighting the potential of obtaining nice triangulations from an input mesh through RoSy field design.