Lasenby describes a Cambridge project with the architecture department that uses drones to study buildings. Since buildings contain many lines, the team wants to process and reconstruct lines from images, where conventional computer vision more often works with matched points.
Applications of Geometric Algebra in Engineering
Speaker
Joan Lasenby, an engineer and researcher at the University of Cambridge, discusses geometric algebra and its engineering applications with a Y Combinator host.
Lasenby traces geometric algebra to Grassmann’s outer product and Clifford’s combination of inner and outer products, later developed as an applied system by David Hestenes. The algebra works with geometric objects such as vectors, oriented planes, and volumes, which can be added, multiplied, and differentiated.
Lasenby explains that geometric algebra offers a way to describe rotations and other geometric operations without relying on stacks of coordinate frames. She says it can make algorithms easier to reason about and program, including camera calibration and motion capture work, though it does not necessarily reduce computation.
Lasenby describes conformal geometric algebra as a five-dimensional framework in which points, lines, planes, circles, and spheres are algebraic objects. Operations such as forming a circle through three points, intersecting shapes, and moving them with rotors fit within the same system.
Lasenby says geometric algebra can help with moving cameras, matching images, and reconstructing lines and planes, while conventional machine learning is used for tasks such as segmentation and recognition. She sees it as a useful toolbox rather than a universal replacement, and points listeners to books, code, and an emerging web-based tool for trying it.
Takeaways
- Use line-based geometry when point clouds do not capture the structure your vision problem needs.
- Choose geometric algebra when rotations, translations, and coordinate frames make an engineering problem hard to express clearly.
- Treat easier reasoning and implementation as potential benefits, not guaranteed reductions in compute.
- Match the method to the task: Lasenby describes machine learning for recognition and geometric algebra for camera motion and reconstruction.
- Try a new mathematical framework on a concrete problem before investing in a broad migration.
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What was claimed 13
- builtDeveloped it as an applied tool and promoted the name geometric algebra.
“David Hestenes came along in the 1960s and said ... 'I'm going to call it geometric algebra.'”
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- featureGeometric algebra: Represents geometric objects and supports operations including addition, multiplication, and differentiation.
“I have an algebra that takes these things as objects, and I can add them, multiply them, and differentiate with respect to them.”
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- builtHis interest in the framework helped introduce Joan Lasenby to it.
“Anthony became fairly obsessed with geometric algebra.”
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- featureQuaternions: Provide a smooth, minimally parametrized way to represent rotations without singularity problems.
“Quaternions are minimally parametrized ... They're smooth and don't suffer from singularity problems.”
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- usesApplies it to engineering and computer vision research.
“I applied to do applications of geometric algebra in engineering.”
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- featureGeometric algebra: Can express rotations and geometric relationships without relying on coordinate matrices.
“I don't have matrices, so I don't have to worry about coordinate systems.”
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- usesJoan Lasenby uses PhaseSpaceWorked on algorithms to calibrate cameras using the company's motion capture system.
“I did some work with a company called PhaseSpace ... looking at algorithms to calibrate cameras.”
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- featureGeometric algebra: Supports calculus on geometric objects and can make analytic differentiation easier.
“Because I have this algebra of objects, I can do calculus on them.”
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