Interactive shape modelling for mathematics visualization

Existing educational virtual spaces simulate either traditional education environments (e.g., lecture theatres, seminar rooms, laboratories) or various phenomena in the real world (e.g., volcano eruptions, tornados, micro and macro scenes). However, subjects richly infused with mathematics and geometry, require from instructors principally new virtual educational spaces, which do not exist in real world. We propose a new approach called to mathematic visualization which emphasizes how complex geometry, colours, textures and other possible properties of virtual shapes can be defined with mathematical formulas and procedures. It immerses the learners into the world of mathematical definitions in explicit, implicit and parametric forms which can be rendered on any suitable graphics system. Since function definitions are small in size compared to traditionally used polygons, the educational environments can be easily used in shared virtual spaces on the internet. To achieve interactivity while working with visual representation of complex mathematical definitions, special measures have to be proposed to accelerate rendering of the shapes defined by implicit functions. We proposed an algorithm for accelerating such rendering for shape modelling where some initial shape defined by an implicit function is gradually modified by other implicitly-defined shapes (tools) with relatively smaller sizes compared to the final shape. It is based on minimization of the volume of the bounding boxes of the tools. The algorithm adds additional mathematical operations to the final function script while the shape is being modelled interactively. Hence, the final definition of the shape is still a function script that can be rendered faster. The proposed method also permits conversion of any originally unstructured function definitions into the accelerated function scripts. As a proof of concept, an educational software tool was implemented for teaching computer graphics and virtual reality to undergraduate and graduate students of Nanyang Technological University.DOCTOR OF PHILOSOPHY (SCE