Numerical recipes in C: the art of scientific computing
Numerical recipes in C: the art of scientific computing
Surface representations of two- and three-dimensional fluid flow topology
VIS '90 Proceedings of the 1st conference on Visualization '90
FAST: a multi-processed environment for visualization of computational fluid dynamics
VIS '90 Proceedings of the 1st conference on Visualization '90
Direct volume visualization of three-dimensional vector fields
VVS '92 Proceedings of the 1992 workshop on Volume visualization
Evaluation of visualization software
ACM SIGGRAPH Computer Graphics - Special focus: modular visualization environments (MVEs)
Image-guided streamline placement
SIGGRAPH '96 Proceedings of the 23rd annual conference on Computer graphics and interactive techniques
Flow visualization for turbomachinery design
Proceedings of the 7th conference on Visualization '96
Visualization of scalar topology for structural enhancement
Proceedings of the conference on Visualization '98
A higher-order method for finding vortex core lines
Proceedings of the conference on Visualization '98
Automatic detection of open and closed separation and attachment lines
Proceedings of the conference on Visualization '98
Interactive deformations from tensor fields
Proceedings of the conference on Visualization '98
Collapsing flow topology using area metrics
VIS '99 Proceedings of the conference on Visualization '99: celebrating ten years
Topology preserving compression of 2D vector fields
Proceedings of the conference on Visualization '00
A topology simplification method for 2D vector fields
Proceedings of the conference on Visualization '00
A flow-guided streamline seeding strategy
Proceedings of the conference on Visualization '00
Locating closed streamlines in 3D vector fields
VISSYM '02 Proceedings of the symposium on Data Visualisation 2002
Quantitative comparative evaluation of 2D vector field visualization methods
Proceedings of the conference on Visualization '01
Continuous topology simplification of planar vector fields
Proceedings of the conference on Visualization '01
IEEE Transactions on Visualization and Computer Graphics
A Predictor-Corrector Technique for Visualizing Unsteady Flow
IEEE Transactions on Visualization and Computer Graphics
The Topology of Symmetric, Second-Order 3D Tensor Fields
IEEE Transactions on Visualization and Computer Graphics
Feature Extraction of Separation and Attachment Lines *
IEEE Transactions on Visualization and Computer Graphics
Tools for Computing Tangent Curves for Linearly Varying Vector Fields over Tetrahedral Domains
IEEE Transactions on Visualization and Computer Graphics
Detection and Visualization of Closed Streamlines in Planar Flows
IEEE Transactions on Visualization and Computer Graphics
IEEE Computer Graphics and Applications
Visualizing Second-Order Tensor Fields with Hyperstreamlines
IEEE Computer Graphics and Applications
Scientific Visualization - Methods and Applications
Informatics - 10 Years Back. 10 Years Ahead.
LIC for Surface Flow Feature Detection
Dagstuhl '97, Scientific Visualization
Computer graphics companion
Iconic Techniques for Feature Visualization
VIS '95 Proceedings of the 6th conference on Visualization '95
Density Functions for Visual Attributes and Effective Partitioning in Graph Visualization
INFOVIS '00 Proceedings of the IEEE Symposium on Information Vizualization 2000
A taxonomy of glyph placement strategies for multidimensional data visualization
Information Visualization
Discrete multiscale vector field decomposition
ACM SIGGRAPH 2003 Papers
Topology Preserving Top-Down Compression of 2D Vector Fields Using Bintree and Triangular Quadtrees
IEEE Transactions on Visualization and Computer Graphics
The virtual windtunnel: an environment for the exploration of three-dimensional unsteady flows
VIS '91 Proceedings of the 2nd conference on Visualization '91
Visualizing wind velocities by advecting cloud textures
VIS '92 Proceedings of the 3rd conference on Visualization '92
Visualization of second order tensor fields and matrix data
VIS '92 Proceedings of the 3rd conference on Visualization '92
Case study: visualizing classical problems in CFD
VIS '92 Proceedings of the 3rd conference on Visualization '92
Visualization of time-dependent flow fields
VIS '93 Proceedings of the 4th conference on Visualization '93
Cloud tracing in convection-diffusion systems
VIS '93 Proceedings of the 4th conference on Visualization '93
The topology of symmetric, second-order tensor fields
VIS '94 Proceedings of the conference on Visualization '94
Differential volume rendering: a fast volume visualization technique for flow animation
VIS '94 Proceedings of the conference on Visualization '94
Visualizing flow over curvilinear grid surfaces using line integral convolution
VIS '94 Proceedings of the conference on Visualization '94
Scientific Visualization: methods and applications
SCCG '03 Proceedings of the 19th spring conference on Computer graphics
Comparing 2D Vector Field Visualization Methods: A User Study
IEEE Transactions on Visualization and Computer Graphics
Visualization of Intricate Flow Structures for Vortex Breakdown Analysis
VIS '04 Proceedings of the conference on Visualization '04
Stream Line and Path Line Oriented Topology for 2D Time-Dependent Vector Fields
VIS '04 Proceedings of the conference on Visualization '04
Tracking of Vector Field Singularities in Unstructured 3D Time-Dependent Datasets
VIS '04 Proceedings of the conference on Visualization '04
Topological Methods for 2D Time-Dependent Vector Fields Based on Stream Lines and Path Lines
IEEE Transactions on Visualization and Computer Graphics
Encyclopedia of Computer Science
Saddle Connectors - An Approach to Visualizing the Topological Skeleton of Complex 3D Vector Fields
Proceedings of the 14th IEEE Visualization 2003 (VIS'03)
Nested OpenMP for efficient computation of 3D critical points in multi-block CFD datasets
Proceedings of the 2006 ACM/IEEE conference on Supercomputing
A Visualization of a Vector-field by a Homogenized Nascent-particles Tracking
Journal of Visualization
Image-Based Streamline Generation and Rendering
IEEE Transactions on Visualization and Computer Graphics
Computation of Localized Flow for Steady and Unsteady Vector Fields and Its Applications
IEEE Transactions on Visualization and Computer Graphics
Stratified helix information of medial-axis-points matching for 3D model retrieval
Proceedings of the international workshop on Workshop on multimedia information retrieval
Computing the curve-skeletons of images
International Journal of Computer Mathematics
Topologically relevant stream surfaces for flow visualization
Proceedings of the 25th Spring Conference on Computer Graphics
ISVC'05 Proceedings of the First international conference on Advances in Visual Computing
A Quantized Boundary Representation of 2D Flows
Computer Graphics Forum
Nearly Recurrent Components in 3D Piecewise Constant Vector Fields
Computer Graphics Forum
Vector field analysis and visualization through variational clustering
EUROVIS'05 Proceedings of the Seventh Joint Eurographics / IEEE VGTC conference on Visualization
Localized flow analysis of 2D and 3D vector fields
EUROVIS'05 Proceedings of the Seventh Joint Eurographics / IEEE VGTC conference on Visualization
Galilean invariant extraction and iconic representation of vortex core lines
EUROVIS'05 Proceedings of the Seventh Joint Eurographics / IEEE VGTC conference on Visualization
Visualization methods for vortex rings and vortex breakdown bubbles
EUROVIS'07 Proceedings of the 9th Joint Eurographics / IEEE VGTC conference on Visualization
Boundary switch connectors for topological visualization of complex 3D vector fields
VISSYM'04 Proceedings of the Sixth Joint Eurographics - IEEE TCVG conference on Visualization
Path line oriented topology for periodic 2D time-dependent vector fields
EUROVIS'06 Proceedings of the Eighth Joint Eurographics / IEEE VGTC conference on Visualization
Topology aware stream surfaces
EuroVis'10 Proceedings of the 12th Eurographics / IEEE - VGTC conference on Visualization
Toward a lagrangian vector field topology
EuroVis'10 Proceedings of the 12th Eurographics / IEEE - VGTC conference on Visualization
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We describe a software system, TOPO, that numerically analyzes and graphically displays topological aspects of a three dimensional vector field, v, to produce a single relatively simple picture that characterizes v. The topology of v that we consider consists of its critical points (where v = 0), their invariant manifolds, and the integral curves connecting these invariant manifolds. Many of the interesting features of v are associated with its critical points. The field in the neighborhood of each critical point is approximated by the Taylor expansion. The coefficients of the first non-zero term of the Taylor expansion around a critical point are the 3x3 matrix ▿v. Critical points are classified by examining ▿v's eigenvalues. The eigenvectors of ▿v span the invariant manifolds of the linearized field around a critical point. Curves integrated from initial points on the eigenvectors a small distance from a critical point connect with other critical points (or the boundary) to complete the topology. In addition, one class of critical surfaces important in computational fluid dynamics is analyzed.TOPO is implemented as a module in the FAST [1] visualization environment. FAST is general purpose visualization software with modules for isosurface generation, particle tracing, etc. TOPO operates on curvilinear, structured grids, including large multi-zone grids. We have used TOPO to visualize a number of computational fluid dynamics (CFD) data sets. The results agree well with other topology software and hand generated topologies. TOPO has proved useful in finding surface topology, flow attachment and separation points, vortex cores, scalar field local extrema, and generally interesting regions of v. We believe there may be other interesting applications yet to be discovered. This paper, along with the references, contains most of the information needed for a scientific programmer to code a topology module in another environment.