Through a New Looking Glass*: Mathematically Precise Visualization
Kirk E. Jordan, Robert M. Kirby, Claudio T. Silva, Thomas Peters · 2010
Alice gazes through her looking glass at a perplexing world that confuses her [3], as vividly depicted in the recently released 3D movie Alice in Wonderland. Scientists, peering through their new looking glass of scientific visualization, derive illuminating insights from otherwise impenetrable petabytes of simulation data. Scientific visualization, however, presents some mathematical challenges of which the general mathematical community should be aware. Most important is a delicate computational balance that must be attained between exact topological properties and their computational approximations [9]. The driving imperative is to prevent visual artifacts that would cloud the scientists ’ interpretations, as happened to a confused Alice. The story of the integration of geometric modeling with computational fluid dynamics and other simulations offers guidance for the solution of similar mathematical problems in scientific visualization. The final stage in the scientific method is the communication of results. This stage is often as important as hypothesis formulation or experimental conclusions. Scientific visualization encompasses both the exploration and the explanation of scientific results. Issues of complexity and precision abound. In contrast to Alice’s world, scientific visualization should produce imagery that illuminates computational experiments. Much as 3D stereovision enriches the movie-viewing experience, mathematically precise visualization will enrich scientific discovery. Scientific visualization relies on many of the same mathematical (numerical) tools used to build simulations, requiring that visualization scientists ask questions concerning model validation and numerical verification. The images in Figure 1, with their increasingly complex geometry and topology, illustrate what’s required: Figure 1(a) has obviously discernible separation at its crossings; as such separations become progressively finer in (b)