3D tensor field design is important in several graphics applications such as procedural noise, solid texturing, and geometry synthesis. Different fields can lead to different visual effects. The topology of a tensor field, such as degenerate tensors, can cause artifacts in these applications. Existing 2D tensor field design systems cannot be used to handle the topology of a 3D tensor field. In this paper, we present to our knowledge the first 3D tensor field design system. At the core of our system is the ability to edit the topology of tensor fields. We demonstrate the power of our design system with applications in solid texturing and geometry synthesis.

Three-dimensional symmetric tensor fields have a wide range of applications in solid and fluid mechanics. Recent advances in the (topological) analysis of 3D symmetric tensor fields focus on degenerate tensors which form curves. In this paper, we introduce a number of feature surfaces, such as neutral surfaces and traceless surfaces, into tensor field analysis, based on the notion of eigenvalue manifold. Neutral surfaces are the boundary between linear tensors and planar tensors, and the traceless surfaces are the boundary between tensors of positive traces and those of negative traces. Degenerate curves, neutral surfaces, and traceless surfaces together form a partition of the eigenvalue manifold, which provides a more complete tensor field analysis than degenerate curves alone. We also extract and visualize the isosurfaces of tensor modes, tensor isotropy, and tensor magnitude, which we have found useful for domain applications in fluid and solid mechanics. Extracting neutral and traceless surfaces using the Marching Tetrahedra method can cause the loss of geometric and topological details, which can lead to false physical interpretation. To robustly extract neutral surfaces and traceless surfaces, we develop a polynomial description of them which enables us to borrow techniques from algebraic surface extraction, a topic well-researched by the computer-aided design (CAD) community as well as the algebraic geometry community. In addition, we adapt the surface extraction technique, called A-patches, to improve the speed of finding degenerate curves. Finally, we apply our analysis to data from solid and fluid mechanics as well as scalar field analysis.

We introduce hexagonal global parameterization, a new type of surface parameterization in which parameter lines respect sixfold rotational symmetries (6-RoSy). Such parameterizations enable the tiling of surfaces with nearly regular hexagonal or triangular patterns, and can be used for triangular remeshing. Our framework to construct a hexagonal parameterization, referred to as HEXCOVER, extends the QUADCOVER algorithm and formulates necessary conditions for hexagonal parameterization. We also provide an algorithm to automatically generate a 6-RoSy field that respects directional and singularity features in the surface. We demonstrate the usefulness of our geometry-aware global parameterization with applications such as surface tiling with nearly regular textures and geometry patterns, as well as triangular and hexagonal remeshing.

Rotational symmetries (RoSys) have found uses in several computer graphics applications, such as global surface parameterization, geometry remeshing, texture and geometry synthesis, and nonphotorealistic visualization of surfaces. The visualization of N-way rotational symmetry (N-RoSy) fields is a challenging problem due to the ambiguities in the N directions represented by an N-way symmetry. We provide an algorithm that allows faithful and interactive representation of N-RoSy fields in the plane and on surfaces, by adapting the well-known line integral convolution (LIC) technique from vector and second-order tensor fields. Our algorithm captures N directions associated with each point in a given field by decomposing the field into multiple different vector fields, generating LIC images of these fields, and then blending the results. To address the loss of contrast caused by the blending of images, we observe that the pixel values in LIC images closely approximate normally distributed random variables. This allows us to use concepts from probability theory to correct the loss of contrast without the need to perform any image analysis at each frame.

This sketch introduces hexagonal global parameterization, a new type of periodic global parameterization that is ideal for tiling surfaces with patterns of six-fold rotational symmetries, i.e., 6-RoSy's [Palacios and Zhang 2007]. Being one of the two most fundamental rotational symmetries that are compatible with translational symmetries in the plane, 6-RoSy's appear in many places in nature, such as honeycombs, insect eyes, corals and crystals, as well as man-made objects such as Islamic patterns [Kaplan and Salesin 2004] and tri-axial weaving [Akleman et al. 2009]. Such symmetries can also provide optimal circle packing, which naturally have applications in architectural design [Schiftner et al. 2009]. A hexagonal global parameterization facilitates all of these applications. See Figure 1 for some examples. Furthermore, parameter lines in a hexagonal global parameterization intersect at an angle of π/3, which enables triangular remeshing of a mesh surface with close to ideal aspect ratios in the triangles. The dual mesh of such a triangulation provides a hexagon-dominant tiling of the surface.

Designing rotational symmetry fields on surfaces is an important task for a wide range of graphics applications. This work introduces a rigorous and practical approach for automatic N-RoSy field design on arbitrary surfaces with user-defined field topologies. The user has full control of the number, positions, and indexes of the singularities (as long as they are compatible with necessary global constraints), the turning numbers of the loops, and is able to edit the field interactively. We formulate N-RoSy field construction as designing a Riemannian metric such that the holonomy along any loop is compatible with the local symmetry of N-RoSy fields. We prove the compatibility condition using discrete parallel transport. The complexity of N-RoSy field design is caused by curvatures. In our work, we propose to simplify the Riemannian metric to make it flat almost everywhere. This approach greatly simplifies the process and improves the flexibility such that it can design N-RoSy fields with single singularity and mixed-RoSy fields. This approach can also be generalized to construct regular remeshing on surfaces. To demonstrate the effectiveness of our approach, we apply our design system to pen-and-ink sketching and geometry remeshing. Furthermore, based on our remeshing results with high global symmetry, we generate Celtic knots on surfaces directly.

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, thetopologyof a rotational symmetry field such assingularitiesandseparatricescan 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 observations thatN-way rotational symmetries can be described by symmetricN-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.