From Pure Maths to shopping and sculpture

Posted by Professor Peter Giblin on 21 March 2016

EPSRC defines impact as research which has social, economic or academic beneficiaries. Pure mathematics underpins ongoing research in a variety of disciplines and in hindsight does not always have a predictable pathway to impact. The unpredictable nature of this type of research can be summarised through the following example of research by Professor Peter Giblin at the University of Liverpool.

Have you ever wanted to try on new clothes in a virtual changing room instead of going to the high street or make a 25 metre high sculpture of a human figure using thousands of metal rods? Well, frankly I haven’t done either of these things; nor did I imagine in the 1980s that fundamental work I was doing on surfaces would ever help shoppers and artists to do them sometime in the future. But ‘pure mathematics’, studied for its own sake - in this case `singularity theory’ - often has a lengthy gestation period. And of course it takes collaborators with a more practical bent to see the possibilities in the maths.

If you hold a ball at arm’s length you see its outline, also known as its profile or apparent contour, as a circle. Naturally, more complex objects have more complex outlines, and as you walk round an object its outline changes. The outline gives a clue to the geometry of the surface, but just how much information does it contain?

In the early 1980s with fellow mathematician, Bill Bruce, I worked on profiles from a purely mathematical point of view (pardon the pun). Then in 1985 I went on leave to the U.S. and ended up working with computer scientist Rich Weiss in the University of Massachusetts at Amherst. Putting together ideas we showed how to use the profiles - gathered from a moving camera - to piece together the surface of the object being viewed when the movement of the camera was known precisely.

But can you get away with knowing only approximately how the camera is moving? Say walking round the object in roughly a circle? Enter another beautiful area of mathematics called the theory of envelopes, together with very classical 19th century projective geometry. Then with research students and other collaborators, including Frank Pollick from California, Roberto Cipolla from Cambridge and Kalle Ǻström from Sweden, it was possible to show that deviations from an exact circle could be corrected for and a highly accurate model made from a sequence of profiles.

This correction required a lot of high-tech numerical work of course, and it has never ceased to amaze me that it actually works. Cipolla and his students at Cambridge have done some amazing things with it, including the ‘Digital Pygmalion’ project to make computer models of sculpture, but also it has been used for online retail (‘Metail®’) and for Anthony Gormley’s astonishing sculpture ‘Exposure’ in the Netherlands.

‘Singularities’ come into all this because when you look at an object, say a ball at arm’s length, then your line of sight grazes the object at points of the circular profile, which is exceptional - singular - behaviour compared with most lines of sight which either hit the object head on or else miss it entirely.

Singularity theory is enormously powerful and has been used for numerous other applications, including optimal planning (‘control theory’), and symmetry and shape segmentation (‘medial representations’ - not medical but there are many medical applications in separating, or segmenting, sections of an computer image of organs in the body). All these came about by applying ideas from pure mathematics studied for their beauty and interest, which were taken up enthusiastically by people who could see applications.

Author

In the following table, contact information relevant to the page. The first column is for visual reference only. Data is in the right column.

Professor Peter Giblin
Name: Professor Peter Giblin
Job title: Professor of Mathematics (Emeritus)
Organisation: University of Liverpool

Peter Giblin is Professor of Mathematics (Emeritus) at the University of Liverpool. He has been a frequent visitor to the United States, where he has taught and collaborated at universities in North Carolina, Massachusetts and Rhode Island.

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