Drawing on Surfaces

Vector graphics in 2D is consolidated since decades, as is supported in many design applications, such as Adobe Illustrator, and languages, like Scalable Vector Graphics (SVG). In this thesis, we address the problem of designing algorithms that support the generation of vector graphics on a discrete surface. We require such algorithm to rely on the intrinsic geometry of the surface, and to support real time interaction on highly-tessellated meshes (few million triangles). Both of these requirements aim at mimicking the behavior of standard drawing systems in the Euclidean context in the following sense. Working in the intrinsic setting means that we consider the surface as our canvas, and any quantity needed to fulfill a given task will be computed directly on it, without resorting to any type of local/global parametrization or projection. In this way, we are sure that, once the theoretical limitations behind some given operation are properly handled, our result will always be consistent with the input regardless of the surface we are working with. As we will see, in some cases, this may imply that one geometric primitive cannot be indefinitely large, but must be contained in a proper subset of the surface. Requiring the algorithms to support real time interaction on large meshes makes possible to use them via a click-and-drag procedure, just as in the $2$D case. Both of these two requirements have several challenges. On the one hand, working with a metric different from the Euclidean one implies that most of the properties on which one relies on the plane are not preserved when considering a surface, so the conditions under which geometric primitives admit a well defined counterpart in the manifold setting need to be carefully investigated in order to ensure the robustness of our algorithms. On the other hand, the building block of most of such algorithms are geodesic paths and distances, which are known to be expensive operations in computer graphics, especially if one is interested in accurate results, which is our case. The purpose of this thesis, is to show how this problem can be addressed fulfilling all the above requirements. The final result will be a Graphical User Interface (GUI) endowed with all the main tools present in a $2$D drawing system that allow the user to generate geometric primitives on a mesh in robust manner and in real-time.