Glyphs for General Second-Order 2D and 3D Tensors

Area of Science:

• Scientific Visualization
• Computational Geometry
• Applied Mathematics

Background:

• Glyphs are crucial for visualizing second-order tensors in scientific data.
• Current methods primarily support symmetric tensors, excluding non-symmetric ones with complex eigenvectors.
• A gap exists in visualizing non-symmetric tensor properties effectively.

Purpose of the Study:

• To develop novel 2D and 3D tensor glyphs capable of representing non-symmetric tensors.
• To ensure these glyphs encode all real eigenvalues and eigenvectors directly.
• To create glyphs with properties like invariance to isometry and scaling, and continuity.

Main Methods:

• Construction of tensor glyphs using piecewise rational curves and surfaces.
• Development of glyphs with properties ensuring a one-to-one mapping between tensors and glyphs.
• Ensuring glyph continuity under tensor variations.

Main Results:

• Successfully created 2D and 3D tensor glyphs based on piecewise rational functions.
• The new glyphs exhibit invariance to isometries and scaling.
• Glyphs directly encode all real eigenvalues and eigenvectors, ensuring a one-to-one tensor-glyph relationship.
• Demonstrated glyph continuity as tensor properties change.

Conclusions:

• The developed tensor glyphs offer a robust method for visualizing both symmetric and non-symmetric second-order tensors.
• These glyphs enhance the representation of physical behavior encoded in geometric properties.
• The application to Jacobian matrix fields in 2D and 3D vector fields demonstrates their utility.