Stuart G. Hoggar

Making Music with Algorithms: A Case-Study System

19 Computer Music Journal, 23:2, pp. 19–30, Summer 1999

64 Lines from a Quaternionic Polytope

An earlier computer calculation produced the vertices of a polytope in quaternionic 4-space, and determined their mutual angles all to be arccos (1/3). This note establishes a computer-independent verification.

Inverse Displacement Mapping

Inverse displacement mapping is a variant of displacement mapping which does not actually perturb the geometry of the surface being mapped. It is thus a true texture mapping technique which can be applied during rendering without breaking viewing pipeline discipline. The method works by first projecting probing rays into texture space and solving for a ray‐texture intersection there. Shadows can also be determined by mapping a probe from the intersection point towards the light source into texture space and seeing if an intersection results. Our implementation uses as much knowledge about the base surface as possible to speed up the ray‐surface intersection calculation. We have limited our treatment to spheres, cones, cylinders and planes, and our rendering method to ray casting, in order to contain the scope of this work up to the present. The inverse displacement mapping technique can, however, be applied more widely, for example as part of a full ray‐tracer, and also as part of the rendering pipeline for a wider class of smooth surfaces.

t -designs with general angle set

The known infinite Delsarte spaces are d -spheres, and projective spaces over the reals, complex numbers, quaternions and octonions. We derive, in a unified manner, intersection numbers and other parameters for general t -designs in these spaces.

Parameters of t-Designs in F P d − 1

We derive the subdegrees for a t-design in projective spaces X = F P d − 1 ( F = ℝ , ℂ , ℍ , O ) . The design may carry an arbitrary finite number of angles. A generalisation of Sidelnikovs inequality is proved and used for this. We give formulae for the parameters of strongly regular graphs that can be realised by a 2 or 3-design carrying two angles, in terms of graph eigenvalues. This leads to a list of feasible parameter sets, some realised.

Tight t -designs and squarefree integers

The authors prove, using a variety of number-theoretical methods, that tight t-designs in the projective spaces FPn of ‘lines’ through the origin in Fn+1 (F = ℂ, or the quarternions H) satisfy t ⩽ 5. Such a design is a generalisation of a combinatorial t-design. It is known that t ⩽ 5 in the cases F = ℝ , O (the octonions) and that t ⩽ 11 for tight spherical t-designs; hence the authors result essentially completes the classification of tight t-designs in compact connected symmetric spaces of rank 1.

Tight 4 and 5-designs in projective spaces

AbstractBannai and Hoggar showed that tightt-designs in projective spaces over ℝ, ℂ, ℍ,

Fractal Compression and the Jigsaw Property I

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