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Creating Polygonal Shapes through a Merger with Topology

Mathematicians at the Institute for Advanced Study in New Jersey have combined two normally separate fields of study, and developed new equations that bridge the gap between them. These new equations relatively simply describe extremely complicated shapes. These include any otherwise simple form that has dynamic alteration along the edge, such as the defect or strain pattern in metal, or exactly how the froth of a wave will fall as it breaks the air.

For fairly obvious reasons, the usefulness of the new math in terms of increasing the reality of highly time-sensitive graphical applications – such as the real-time rendering of interactive VR – cannot be overstated.

This new work is described in the American Institute of Physics' Journal of Mathematical Physics (JMP), and is open access to the general public. In the work, all new equations and their effects are detailed at length. The work is also highly useful for taking a simple shape, punching a hole in it, and with a single, relatively simple equation, working out the precise shape and area of the destroyed area. This has obvious implications for infinitely destructible areas, and the real-time on-the-fly modification of models without resorting to labour-intensive number crunching (which cannot be performed in real-time).

The work relies upon a recently developed mathematical theory, known as "persistent homology," which takes into account the sizes and number of holes in a geometric shape. The work described in JMP is a proof of concept based on fractals that have already been studied by other methods -- such as the shapes assumed by large polymer molecules as they twist or bend under random thermal fluctuation.

The mathematicians plan to use the vocabulary provided by persistent homology methods to investigate and describe complicated shapes in a whole new way, which we will be keeping a close eye upon.

Three examples of the internal structure of a shape, formed through different processes as described in this work:
(a) Branched polymer,
(b) Brownian tree
(c) self-avoiding walk.

The real advantage here is that any shape can be modified according to one of the three patterns, and its area superimposed on one of the possible fractal patterns, at a known pointy or line on its surface, with a simple equation. So for example, a self-avoiding walk computed as the uppermost vector of a wave-model, produces a perfect frothy surf ridge, unique to that wave, and without straining the graphics engine's capabilities.


Mathematicians Develop New Method for Describing Extremely Complicated Shapes

Measuring shape with topology @ Journal of Mathematical Physics

Measuring shape with topology (PDF)

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