Stripe Parameterization of Tubular Surfaces

We present a novel algorithm for automatic parameterization of tubelike surfaces of arbitrary genus, such as the surfaces of knots, trees, blood vessels, neurons, or any tubular graph with a globally consistent stripe texture. Mathematically, these surfaces can be described as thickened graphs, and the calculated parameterization stripe will follow either around the tube, along the underlying graph, a spiraling combination of both, or obey an arbitrary texture map whose charts have a 180 degree symmetry. We use the principal curvature frame field of the underlying tube-like surface to guide the creation of a global, topologically consistent stripe parameterization of the surface. Our algorithm extends the QuadCover algorithm and is based, first, on the use of so-called projective vector fields instead of frame fields, and second, on different types of branch points. That does not only simplify the mathematical theory, but also reduces computation time by the decomposition of the underlying stiffness matrices.