YC
Joan Lasenby on Applications of Geometric Algebra in Engineering
@ Y Combinator
09/17/18
Takeaways
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Joan Lasenby
09/17/18
@ Y Combinator
We're using drones to analyze the built environment, focusing on processing lines rather than points, which is traditionally more challenging in computer vision.
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Joan Lasenby
09/17/18
@ Y Combinator
Geometric algebra provides a mathematical framework that allows us to work with lines in computer vision, which is more complex than working with points.
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Joan Lasenby
09/17/18
@ Y Combinator
David Hestenes revived interest in geometric algebra in the 1960s, recognizing its potential for simplifying complex mathematical concepts.
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Joan Lasenby
09/17/18
@ Y Combinator
Geometric algebra allows for a unified approach to various fields, enabling work in rigid body dynamics, linear algebra, and quantum physics without relying on matrices or tensors.
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Joan Lasenby
09/17/18
@ Y Combinator
The challenge in adopting geometric algebra lies in the established use of vector calculus and linear algebra, which makes it difficult for researchers to shift to this new mathematical system.
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Joan Lasenby
09/17/18
@ Y Combinator
My husband, Anthony, became fascinated with geometric algebra, which led me to explore its applications in engineering, particularly in rotations.
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Joan Lasenby
09/17/18
@ Y Combinator
In engineering, the way geometric algebra handles rotations is crucial, as it simplifies the complexities associated with traditional rotation matrices.
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Joan Lasenby
09/17/18
@ Y Combinator
Quaternions are preferred for rotations in three dimensions because they are minimally parametrized and avoid singularity issues, making them smoother to work with than rotation matrices.
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Joan Lasenby
09/17/18
@ Y Combinator
In 1993, the state of computer vision was primarily focused on geometry, lacking the machine learning techniques that are prevalent today. In the early 90s, Bayesian statistics began to influence computer vision, enabling better tracking of objects in images and identifying the most probable paths, especially in crowded environments.
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Joan Lasenby
09/17/18
@ Y Combinator
I was fortunate to receive a five-year postdoctoral fellowship from the Royal Society to explore applications of geometric algebra in engineering, allowing me to focus on my research with minimal teaching responsibilities.
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Joan Lasenby
09/17/18
@ Y Combinator
One of my early projects involved translating classical projective geometry concepts into geometric algebra, which simplified the mathematical complexity of computer vision tasks.
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Joan Lasenby
09/17/18
@ Y Combinator
Using geometric algebra provides a clear origin point for calculations, eliminating the confusion of multiple coordinate systems and making it easier to handle rotations and translations in computer vision.
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Joan Lasenby
09/17/18
@ Y Combinator
The goal during my five-year research period was to apply geometric algebra to practical uses and basic research, exploring its potential in various applications.
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Joan Lasenby
09/17/18
@ Y Combinator
I collaborated with a motion capture company to develop algorithms for camera calibration, leveraging the unique properties of geometric algebra to enhance the process.
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Joan Lasenby
09/17/18
@ Y Combinator
Geometric algebra allows for easy differentiation of geometric objects, making it simpler to derive solutions compared to traditional methods involving matrices or tensors.
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Joan Lasenby
09/17/18
@ Y Combinator
The beauty of geometric algebra lies in its ability to represent complex geometric transformations intuitively, making programming and computation more straightforward.
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Joan Lasenby
09/17/18
@ Y Combinator
Conformal geometric algebra, which extends to five-dimensional space, allows for the representation of points, lines, planes, and spheres as algebraic objects, simplifying many geometric operations.
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Joan Lasenby
09/17/18
@ Y Combinator
In modern computer vision, there's a significant shift from geometry to machine learning, where algorithms learn to recognize and segment objects from vast amounts of image data.
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Joan Lasenby
09/17/18
@ Y Combinator
While geometric algebra may not provide new capabilities beyond conventional methods, it offers a unique perspective that can inspire innovative approaches to solving geometric problems.
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Joan Lasenby
09/17/18
@ Y Combinator
In my research, we are keen on analyzing moving images to extract geometric objects and understand motion, which is crucial for applications like self-driving cars. Geometric algebra serves as a unifying language, simplifying complex computations in fields like computer vision, making it easier to work with concepts like rotations and translations without confusion.
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Joan Lasenby
09/17/18
@ Y Combinator
The strength of geometric algebra lies in its ability to unify various concepts in physics, including quantum mechanics and relativity, potentially offering new insights into quantum gravity.
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Joan Lasenby
09/17/18
@ Y Combinator
Geometric algebra provides a more intuitive approach to complex mathematical concepts, allowing individuals with geometric insight but less mathematical sophistication to engage with advanced topics effectively.
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Joan Lasenby
09/17/18
@ Y Combinator
There are numerous resources available for learning geometric algebra, including books by David Hestenes and others, as well as online tools and packages that facilitate its application in programming environments like Python and MATLAB.
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Joan Lasenby
09/17/18
@ Y Combinator
A web version of geometric algebra tools is being developed to make it more accessible, allowing users to experiment without the need for complex installations or configurations.