Topological analysis for 3D real, symmetric second-order tensor fields using Deviatoric Eigenvalue Wheel

Abstract

Tensor fields are of particular interest not solely because they represent a wide variety of physical phenomena but also because crucial importance can be inferred from vector and scalar fields in terms of the gradient and Hessian, respectively. This work presents our key insights into the topological structure of 3D real, symmetric second-order tensor fields, of which the central goal is to show a novel computation model we call Deviatoric Eigenvalue Wheel. Based on the computation model, we show how the eigenanalysis for tensor fields can be simplified and put forward a new categorizer for topological lines. Both findings outperform existing approaches as our computation depends on the tensor entries only. We finally make a strict mathematical proof and draw a conclusion of R=Kcos(3α), wherein R is the tensor invariant of determinant, K is a tensor constant, and α is called Nickalls angle. This conclusion allows users to better understand how the feature lines are formed and how the space is divided by topological structures. We test the effectiveness of our findings with real and analytic tensor fields as well as simulation vector fields.