Geometry in Engineering: Q&A with Professor Joan Lasenby

Professor Joan Lasenby was among the first female undergraduates at Trinity in 1978. She studied Mathematics and admits that at the time ‘engineering was something of a mystery to me’. She went on to a PhD in Radio and Molecular Studies at the Cavendish Laboratory.

As well as posts at Louisiana State University, Trinity Hall and Newnham College, Cambridge, Professor Lasenby has worked in industry. She was elected a Fellow of Trinity in 2001, where she is Director of Studies in Engineering. She is Professor of Image and Signal Analysis at Cambridge. At the third Women of Trinity Lecture, Geometry in Engineering, on Wednesday 28 November, Professor Lasenby considered the range of applications of geometry – for example in computer vision, computer graphics and structural engineering. She has a particular interest in geometric algebra and here she explains why.

First of all, what is geometric algebra? 

One should think of geometric algebra as a tool – a unifying mathematical framework which enables you to tackle many problems in engineering and physics. These problems can, of course, often be tackled by other means, but geometric algebra sometimes give insights that would be hard to get otherwise, due to the geometric intuition it brings.

What can be gained from studying geometric algebra and what career paths does it open up? 

In today’s world we have ever-increasing amounts of data and ever more advanced sensing devices. For example, drones can now fly into areas that may be too dangerous for humans. Being able to remotely monitor the path of the drone and to accurately build up a 3D map of the scanned terrain are important functions. The computer vision techniques that enable us to do this are based on geometric reasoning.

In the job market today, the most sought-after skill is expertise in Machine Learning(ML)/Artifical Intelligence(AI). Platforms are now available which enable complex ML architectures to be easily constructed. In computer vision, the most basic task is to determine the camera positions given matching points in multiple views – in today’s state of the art systems, this task can increasingly be accomplished by ML. In tomorrow’s job market, people who have basic geometry skills as well as ML expertise will be those who push forward research in this field.

Can you give an example of where a geometric approach to a problem has made progress possible?

If one is matching objects, for example a line in the world projected into two different camera views or 3D world lines in robotic analysis, a common question is ‘how close is one line to another?’ This is not a straightforward question. Indeed, one can also ask, how close is one plane to another, or one circle to another etc? Building up a sensible measure of such ‘differences’ requires geometric approaches, and in particular, geometric techniques which are covariant (meaning it doesn’t matter where in space it is, you will still get the same answer).

Why was Trinity Fellow William Kingdon Clifford (1868-1871) important?

Clifford was an amazing mathematician and philosopher, making contributions in many fields. His untimely death at the age of just 34 did not help the dissemination of his work. At the time that Clifford produced his ‘Geometric Algebra’, Gibbs et al introduced ‘vector calculus’, which gained huge popularity, partly because of the beautiful way it handled eletromagnetism and Maxwell’s equations. Clifford’s algebra was then mostly forgotten by applied scientists. In the 1960s David Hestenes revived geometric algebra (the name Clifford gave it) and made some astonishing discoveries, using it to simplify Maxwell’s equations, the Pauli matrices, and rigid body dynamics etc.

Listen to a podcast of Professor Lasenby’s lecture.