Using geometric algebra to interactively model the geometry of Euclidean and non-Euclidean spaces.

This research interprets and develops the 'conformal model of space' in a way appropriate for a graphics developer interested in the design of interactive software for exploring 2-dimensional non-Euclidean spaces. The conformal model of space extends the standard projective model – instead of adding just one extra dimension to standard Euclidean space, a second one is added that results in a Minkowski space similar to that of relativistic spacetime. Also, standard matrix algebra is replaced by geometric ( i.e. Clifford) algebra. The key advantage of the conformal model is that both Euclidean and non- Euclidean spaces are accommodated within it. Transformations in conformal space are generated by bivectors which are special elements of the geometric algebra. These induce geometric transformations in the embedded non Euclidean spaces. However, the relationship between the bivector generated transformations of the Minkowski modelling space and the geometric transformations they induce is extremely obscure. This thesis provides new analytical tools for determining the nature of this relationship. Their derivation was motivated by the need to successfully solve key implementation problems relating to navigation and in-scene mouse interaction. The analytic approaches developed not only successfully solved these problems but pointed the way to implementing other unplanned features. These include facilities for dynamically altering on-screen geometry as well as using multiple viewports to allow the user to interact with the same objects embedded in different geometries. These new analytical approaches could be powerful tools for solving future and as yet unforeseen implementation problems

Using geometric algebra to interactively model the geometry of Euclidean and non-Euclidean spaces

This research interprets and develops the 'conformal model of space' in a way appropriate for a graphics developer interested in the design of interactive software for exploring 2-dimensional non-Euclidean spaces. The conformal model of space extends the standard projective model – instead of adding just one extra dimension to standard Euclidean space, a second one is added that results in a Minkowski space similar to that of relativistic spacetime. Also, standard matrix algebra is replaced by geometric ( i.e. Clifford) algebra. The key advantage of the conformal model is that both Euclidean and non- Euclidean spaces are accommodated within it. Transformations in conformal space are generated by bivectors which are special elements of the geometric algebra. These induce geometric transformations in the embedded non Euclidean spaces. However, the relationship between the bivector generated transformations of the Minkowski modelling space and the geometric transformations they induce is extremely obscure. This thesis provides new analytical tools for determining the nature of this relationship. Their derivation was motivated by the need to successfully solve key implementation problems relating to navigation and in-scene mouse interaction. The analytic approaches developed not only successfully solved these problems but pointed the way to implementing other unplanned features. These include facilities for dynamically altering on-screen geometry as well as using multiple viewports to allow the user to interact with the same objects embedded in different geometries. These new analytical approaches could be powerful tools for solving future and as yet unforeseen implementation problems.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Articulating Space: Geometric Algebra for Parametric Design -- Symmetry, Kinematics, and Curvature

To advance the use of geometric algebra in practice, we develop computational methods for parameterizing spatial structures with the conformal model. Three discrete parameterizations – symmetric, kinematic, and curvilinear – are employed to generate space groups, linkage mechanisms, and rationalized surfaces. In the process we illustrate techniques that directly benefit from the underlying mathematics, and demonstrate how they might be applied to various scenarios. Each technique engages the versor – as opposed to matrix – representation of transformations, which allows for structure-preserving operations on geometric primitives. This covariant methodology facilitates constructive design through geometric reasoning: incidence and movement are expressed in terms of spatial variables such as lines, circles and spheres. In addition to providing a toolset for generating forms and transformations in computer graphics, the resulting expressions could be used in the design and fabrication of machine parts, tensegrity systems, robot manipulators, deployable structures, and freeform architectures. Building upon existing algorithms, these methods participate in the advancement of geometric thinking, developing an intuitive spatial articulation that can be creatively applied across disciplines, ranging from time-based media to mechanical and structural engineering, or reformulated in higher dimensions