Splines for engineers: with selected applications in numerical methods and computer graphics

Splines are piecewise polynomial, rational or trigonometric functions that are ubiquitous in a wide variety of areas. Popular applications include function approximation as well as parametric curves and surfaces. In this dissertation, we study them from the perspective of both numerical methods and computer graphics. A pragmatic and richly illustrated approach is used to explain the different subject matters.

Following a refresher on spline basics, we consider a spline-based subdivision scheme for three-valent meshes (that is, meshes composed of mostly hexagons). This provides us with a good understanding of subdivision surfaces in general, a modelling technique used in animated movies, VFX and computer games. We then move on to study improved quadrature (that is, numerical integration) for the widely used Catmull-Clark subdivision scheme for quad-dominant meshes, something keenly awaited by spline-based numerical methods such as isogeometric analysis (IgA). These methods deal with the numerical simulation of all sorts of physical phenomena, including heat conduction and (elastic) deformation. Finally, we shift our focus to a more artistic use of splines in the context of vector graphics. Here, we propose user-friendly additions such as local refinement for the gradient mesh primitive, facilitating the creation of resolution-independent (almost) photo-realistic illustrations.