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We present an algorithm for the simulation of incompressible fluid phenomena that is computationally efficient and leads to visually convincing simulations with far fewer degrees of freedom than existing approaches. Rather than using an Eulerian grid or Lagrangian elements, we represent vorticity and velocity using a basis of global functions defined over the entire simulation domain. We show that choosing Laplacian eigenfunctions for this basis provides benefits, including correspondence with spatial scales of vorticity and precise energy control at each scale. We perform Galerkin projection of the Navier-Stokes equations to derive a time evolution equation in the space of basis coefficients. Our method admits closed form solutions on simple domains but can also be implemented efficiently on arbitrary meshes.

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We propose a conceptual extension of the standard triangle-based graphics pipeline by an additional intersection stage. The corresponding intersection program performs ray-object intersection tests for each fragment of an object's bounding volume. The resulting hit fragments are transfered to the fragment shading stage for computing the illumination and performing further fragment operations. Our approach combines the efficiency of the standard hardware graphics pipeline with the advantages of ray casting such as pixel accurate rendering and exact normals as well as early ray termination.This concept serves as a framework for the implementation of an interactive ray casting system for trimmed NURBS surfaces. We show how to realize an iterative ray-object intersection method for NURBS primitives as an intersection program. Convex hulls are used as tight bounding volumes for the NURBS patches to minimize the number of fragments to be processed. In addition, we developed a trimming algorithm for the GPU that works with an exact representation of the trimming curves. First experiments with our implementation show that real-time rendering of medium complex scenes is possible on current graphics hardware.

Spherical harmonics are employed in a wide range of applications in computational science and physics, and many of them require the rotation of functions. We present an efficient and accurate algorithm for the rotation of finite spherical harmonics expansions. Exploiting the pointwise action of the rotation group on functions on the sphere, we obtain the spherical harmonics expansion of a rotated signal from function values at rotated sampling points. The number of sampling points and their location permits one to balance performance and accuracy, making our technique well-suited for a wide range of applications. Numerical experiments comparing different sampling schemes and various techniques from the literature are presented, making this the first thorough evaluation of spherical harmonics rotation algorithms.