ITR: A Unified Representation for Non-homogeneous Models Manifesting Surface, Volume, and Vector Attributes

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This research project will develop improved mathematical representations of three-dimensional objects, for use across a variety of applications. Underlying model representations in computer graphics, computer-aided geometric design, engineering analysis, and medical visualization have historically been driven by different requirements. Thus, different algorithmic and computational foundations have evolved yielding two largely disparate formulations. Differences in model representations impede cross-pollination of valuable techniques, hinder model sharing, and limit cross-area applications. To attack this historically intractable problem, a unified mathematical representation that can work across a broad spectrum of applications will be developed, based on an extended, multidimensional, parametric tensor product B-spline hyper-volume. High dimensional NURBs (Non-Uniform Rational B-Splines) are used as a basis because of their ubiquity in practice, universality in modeling, good overall computational and interactive behavior, and extensive understanding. Distinguishing itself from the straightforward generalization, the extended representation will support a non-hyperrectangular domain (non-hexahedral for the trivariate) appropriate to the model, and include supplementary attributes, defined over a common multidimensional parametric domain. The attributes will model both functional characteristics and data represented as scalars or vectors.Recent advances in several key technologies are creating demand for higher dimensional representations and corresponding visualization algorithms to design and analyze truly solid geometry, i.e., working in a design space of non-homogeneous materials manifesting multidimensional attributes. This calls for fundamental advances in our computing and visualization environments. One of the practical applications is designing optical lenses that achieve refracting power by smoothly varying the index of refraction and keeping a simple geometric shape. The combination of simple shape with no discontinuities in the index of refraction offers substantial optical and mechanical advantages over traditional multifarious lens systems. The project will also have educational benefits, training students in the laboratory to help them become leaders in this increasingly important field.