Vector field design on surfaces

Abstract

Vector field design on surfaces is necessary for many graphics applications: example-based texture synthesis, nonphotorealistic rendering, and fluid simulation. For these applications, singularities contained in the input vector field often cause visual artifacts. In this article, we present a vector field design system that allows the user to create a wide variety of vector fields with control over vector field topology, such as the number and location of singularities. Our system combines basis vector fields to make an initial vector field that meets user specifications.The initial vector field often contains unwanted singularities. Such singularities cannot always be eliminated due to the Poincaré-Hopf index theorem. To reduce the visual artifacts caused by these singularities, our system allows the user to move a singularity to a more favorable location or to cancel a pair of singularities. These operations offer topological guarantees for the vector field in that they only affect user-specified singularities. We develop efficient implementations of these operations based on Conley index theory. Our system also provides other editing operations so that the user may change the topological and geometric characteristics of the vector field.To create continuous vector fields on curved surfaces represented as meshes, we make use of the ideas of geodesic polar maps and parallel transport to interpolate vector values defined at the vertices of the mesh. We also use geodesic polar maps and parallel transport to create basis vector fields on surfaces that meet the user specifications. These techniques enable our vector field design system to work for both planar domains and curved surfaces.We demonstrate our vector field design system for several applications: example-based texture synthesis, painterly rendering of images, and pencil sketch illustrations of smooth surfaces. Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or direct commercial advantage and that copies show this notice on the first page or initial screen of a display along with the full citation. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, to republish, to post on servers, to redistribute to lists, or to use any component of this work in other works requires prior specific permission and/or a fee. Permissions may be requested from Publications Dept., ACM, Inc., 2 Penn Plaza, Suite 701, New York, NY 10121-0701 USA, fax +1 ( . In nonphotorealistic rendering, vector fields are used to guide the orientation of brush strokes [Hertzmann 1998] and hatches [Hertzmann and Zorin 2000]. In fluid simulation, the external force is a vector field which need not correspond to any physical phenomenon and can exist on synthetic 3D surfaces [Stam 2003]. A vector field design system enables these applications to achieve many different visual effe...

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