GA Robotics
Interviewee: Joan Lasenby, Professor of Image and Signal Analysis, Department of Engineering, University of Cambridge
Your work spanned several domains — motion capture, computer vision, and machine learning. Across these fields, where does Geometric Algebra play the most central role?
The clearest example is probably motion capture, where the entire pipeline is inherently geometric. The process begins with calibrating a camera system by optimising a cost function to determine camera positions. From there, one tracks markers in space, reconstructs a skeletal structure. Then, if you want rigging and skinning, we have a process named “bone glow”, which is basically equivalent to dual quaternions.
This effectively formed a complete geometric framework from raw capture data to animated output.
That emphasis on optimisation seems closely related to machine learning. How does Geometric Algebra translate into that context?
In geometric approaches to neural networks, inputs, weights, and biases can all be represented as multi-vectors or any structure you wish. The key is to ensure that operations preserve covariance — for instance, using sandwich products so that transformations applied to inputs propagate consistently through the network.
However, current implementations are still limited: although the representations are geometric, optimisation is typically performed by decomposing everything into scalars and then apply back-propagation, which is hardly ideal. The other thing that we did was to re-parametrise our problem and the end-cost function in terms of Geometric Algebra.
I believe that the whole optimisation pipeline can be probably improved by leaving the GA objects as they are and do the differentiation.
You also suggested that Geometric Algebra could replace parts of classical calculus. What makes it more powerful?
In standard calculus, differentiation is usually defined with respect to scalars or vector components. Geometric Algebra allows me to differentiate with respect to any multi-vector. It is not something you sort of learn at the beginning, but it is very powerful. You can for instance differentiate with respect to a rotor, which is a technique I use all the time.
Despite these advantages, Geometric Algebra is not widely taught at undergraduate level. Why is that?
The issue is less about students and more about structure. Curricula are already dense, so introducing GA requires replacing something else, which means convincing colleagues. That’s often the real challenge.
Interestingly, students tend to grasp GA quite quickly. The barrier is more institutional than intellectual.
When making the case for Geometric Algebra, what arguments do you find most convincing?
A classic example, which was also brought up during Freya Holmèr’s talk, is the cross product. It only exists in three dimensions and leads to somewhat artificial distinctions, like vectors versus pseudo-vectors. In GA, these issues disappear by teaching students from the beginning that pseudo-vectors are planes.
Starting from that perspective simplifies many concepts that are otherwise taught in a more fragmented way.
Looking at your broader research, what project are you particularly proud of?
It’s difficult to single out one, but work on fabric simulation has had a notable influence. However, people still don’t use constraints. I think we should create an open-source package where constraint usage is more straightforward.
Are there current projects you’re especially excited about?
Yes, one involves an open-source package in Julia that explores alternative Geometric Algebras, such as Sphere Geometry. Different algebras lend themselves to different applications. This one allows you to work with angle constraints, for example enforcing that objects meet at specific angles.
It’s an area where experimentation is essential, especially since the research landscape is still relatively sparse.
When working in such under–explored areas, how do you proceed?
You must build tools that let you experiment. If there is not much prior work, the only way forward is to try things yourself.
Finally, how do you see the future of Geometric Algebra in industry and research?
It will be adopted; it is only a matter of when that will be the case. Progress has been slower than one might hope, partly because researchers tend to focus on their own research.
That said, there’s reason for optimism. Game development, for instance, was quick to adopt quaternions for rotations, even though they had existed for a long time. A similar path could emerge for GA. If a compelling application appears, perhaps in games or real-time graphics, adoption could accelerate and spread to other fields.
Interview conducted by Bas Feitsma