Taking shape: computational geometry

21 May 2017

Geometry is the theoretical study of shapes, while computational geometry aims to establish algorithms that use ideas of geometry to tackle real-life problems. This is the area of research which has gripped Mr Gary Pui-Tung Choi (2016 Croucher Scholar).

It was during his BSc and MPhil, both in Mathematics, that Choi learned of and subsequently developed an interest in computational geometry, and began developing geometric algorithms to be applied to computer graphics. His work includes surface registration and texture mapping, with an aim of improving graphical effects. However, during the second year of his MPhil, he felt that he might like to diversify his research. Choi now studies for an Applied Mathematics PhD in the John A. Paulson School of Engineering and Applied Sciences at Harvard University, although the focus of the applications of his research has shifted from computer graphics to the fields of physics, biology and medicine, while still addressing geometric problems.

Application in physics setting

One of the applications of this research in a physics setting involves designing structures which can grow when stretched. For instance, Choi would begin with a piece of paper and introduce suitable cuts which try to expand the surface and create a new shape. His study explores how different cuts can create different shapes, ultimately designing a geometric algorithm that could automatically determine the cuts needed for any given final shape. Choi also made it clear that this was only the beginning of this research, highlighting that, “If we can study this paper-cutting problem, it may be useful for the design of some materials in engineering and arts.” He also stated that, for example, “A potential application of this work is the manufacturing of membrane filters using the physical property of auxetic tessellation.”

Application in biological setting

Meanwhile, the biological problems Choi has been addressing involve examining Drosophila wings, the wings of “fruit flies”, the type of wings found in insects, and analysing how the shapes of these wings vary with different species. This allows him to explore the relationship between the shapes of wings and genetics. Using quasi-conformal mappings to focus on the shape and vein structure of Drosophila wings, Choi has been able to study how these shapes differ from one another, providing more understanding of the development and evolution processes of Drosophila wings. Choi has also developed a quantitative method to compare different wings and evaluate any differences. This method has allowed him to accurately compare the vein structure and pigmentation of Drosophila wings, and to classify the wings based on their phenotypic features.

Study of nectar spurs development

Another biological problem Choi is addressing involves the study of nectar spurs development in certain flowers in collaboration with the Department of Organismic and Evolutionary Biology at Harvard. Biologists have discovered that crossing certain species of flowers produces specific spur patterns, creating a large variety of different shapes, including longer or shorter spurs or the absence of spurs. This suggests that there is a genetic control on spur development. Choi is now focused on developing a geometric method to evaluate differing curvatures of petal nectar spurs, and to analyse their relationship with genetic variations. He ultimately aims to relate this to the plant’s genetic information.

Application within the medical field

By gaining a better understanding of these relatively simple shapes, Choi hopes to then be able to study more complicated shapes, such as the folding within the human brain. In another collaborative effort, this time with the Harvard Medical School, he is attempting to simulate the cortical convolution of the brain using both computational model and physical gel model. The aim is to explore whether the simulated folding patterns match medical data, using curvature-based surface mapping techniques. This could potentially have real-life applications within the medical field, with Gary highlighting that, “We know that there are a lot of diseases which are related to the brain, so if we can understand the brain in a more quantitative way, then that may give us a way of measuring and comparing different brains, such as comparing normal brains with patient’s, such as Alzheimer’s disease.”

While so far computational geometry has mostly been applied to graphics and engineering fields, Choi aims to promote computational geometry in a variety of different fields, as he believes that there are a large number of potential applications for this work, in biology, physics, medicine and other scientific areas. He hopes to develop geometry techniques to tackle various different applications. Choi also aspires to use his expertise in computational geometry to contribute to Hong Kong in some way, be it as a physics, biological, medical, or other type of application.

Mr Choi obtained his BSc (First Class Honours) in 2014 and MPhil in 2016, both in Mathematics and from the Chinese University of Hong Kong. In 2016 he was awarded a Croucher Scholarship and became a PhD student in the John A. Paulson School of Engineering and Applied Sciences at Harvard University, studying applied mathematics. His research focuses on computational differential geometry and geometric morphometrics, taking both biological and physical approaches.

To view Mr Choi’s Croucher profile, please click here