  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.
