Dr Rich Wareham

This paper presents a method of interpolating between two or more general displacements (rotation and translation). The resulting interpolated path is smooth and possesses a number of desirable properties. It differs from existing algorithms which require factorising the pose into separate rotation and translation components and is derived from an intuitively appealing framework-i.e. a natural extension of the standard interpolation scheme for pure rotations. While this paper presents the theory behind the interpolation and its description as a tool, we also outline the possible advantages of using this technique for vision and graphics applications.

Many real-time algorithms for mesh deformation driven by animation of an underlying skeleton make use of a set of per-bone weights associated with each vertex. There are few unguided algorithms for the assignment of these weights with a recent proposed solution being bone heat [?]. In this paper, we briefly discuss bone heat and provide examples where it performs poorly. We then develop a refinement of bone heat, termed bone glow, which, in our validation, performed as well as bone heat in simple cases while not suffering from bone heat's observed weaknesses.

This paper introduces the mathematical framework of conformal geometric algebra (CGA) as a language for computer graphics and computer vision. Specifically it discusses a new method for pose and position interpolation based on CGA which firstly allows for exisiting interpolation methods to be cleanly extended to pose and position interpolation, but also allows for this to be extended to higher-dimension spaces and all conformal transforms (including dilations). In addition, we discuss a method of dealing with conics in CGA and the intersection and reflections of rays with such conic surfaces. Possible applications for these algorithms are also discussed.

Using the algebra developed by Clifford (1878) from the work of Graßmann (1877) and the mapping introduced by Hestenes & Sobczyk (1984), which we call the conformal model, certain geometric objects (e.g. spheres, lines, circles, etc) may be conveniently represented as multivectors in a higher dimension space. Furthermore, geometric operations on these objects (e.g. intersections, rotations, inversions etc.) may be performed efficiently and intuitively within this space. Also the model generalizes well to certain non-Euclidean geometries. A prototype implementation of this model has been created and this paper will describe it along with some of the approaches taken to reduce the computational complexity required. The possibility of implementation in hardware will be discussed and example output will be presented.

Under the mapping from R³ to R⁵ introduced by Hestenes, which we call the conformal model, certain geometric objects in R³ (e.g. spheres, lines, circles, etc) may be conveniently represented as blades in R⁵. Furthermore, geometric operations on these objects (e.g. intersections, rotations, inversions etc.) may be performed efficiently and intuitively within this space. Due to its intuitive nature, this approach lends itself to problems where speed of implementation is important or problems where the complexity quickly leads to large numbers of `special case conditions' within classical Vector Algebra. A prototype implementation has been created and this project will describe it along with some of the approaches taken to reduce the computational complexity required. The possibility of implementation in hardware will be discussed.