MVF Designer: Design and Visualization of Morse Vector Fields

TL;DR

MVF Designer is an interactive system that enables mathematicians to design and visualize Morse vector fields and Morse-Smale complexes, facilitating exploration of complex topological configuration spaces.

Contribution

The paper introduces MVF Designer, a novel interactive tool for designing and visualizing Morse vector fields and their topological structures, with fine-grained control and elementary editing moves.

Findings

Enables exploration of Morse function configuration spaces.

Supports detailed control over vector field topology.

Facilitates applications in topological data analysis.

Abstract

Vector field design on surfaces was originally motivated by applications in graphics such as texture synthesis and rendering. In this paper, we consider the idea of vector field design with a new motivation from computational topology. We are interested in designing and visualizing vector fields to aid the study of Morse functions, Morse vector fields, and Morse-Smale complexes. To achieve such a goal, we present MVF Designer, a new interactive design system that provides fine-grained control over vector field geometry, enables the editing of vector field topology, and supports a design process in a simple and efficient way using elementary moves, which are actions that initiate or advance our design process. Our system allows mathematicians to explore the complex configuration spaces of Morse functions, their gradients, and their associated Morse-Smale complexes. Understanding these spaces will help us expand further their applicability in topological data analysis and visualization.