A 1d Up Approach to Conformal Geometric Algebra: Applications in Line Fitting and Quantum Mechanics

@article{Lasenby2020A1U,
  title={A 1d Up Approach to Conformal Geometric Algebra: Applications in Line Fitting and Quantum Mechanics},
  author={Anthony N. Lasenby},
  journal={Advances in Applied Clifford Algebras},
  year={2020},
  volume={30},
  pages={1-16}
}
  • A. Lasenby
  • Published 22 February 2020
  • Mathematics
  • Advances in Applied Clifford Algebras
We discuss an alternative approach to the conformal geometric algebra (CGA) in which just a single extra dimension is necessary, as compared to the two normally used. This is made possible by working in a constant curvature background space, rather than the usual Euclidean space. A possible benefit, which is explored here, is that it is possible to define cost functions for geometric object matching in computer vision that are fully covariant, in particular invariant under both rotations and… 
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References

SHOWING 1-10 OF 10 REFERENCES
Calculating the Rotor Between Conformal Objects
In this paper we will address the problem of recovering covariant transformations between objects—specifically; lines, planes, circles, spheres and point pairs. Using the covariant language of
Rigid Body Dynamics in a Constant Curvature Space and the '1D-up' Approach to Conformal Geometric Algebra
  • A. Lasenby
  • Mathematics
    Guide to Geometric Algebra in Practice
  • 2011
TLDR
A ‘1D up’ approach to Conformal Geometric Algebra, which treats the dynamics of rigid bodies in 3D spaces with constant curvature via a 4D conformal representation, and the final view of ordinary translational motion is shown to correspond to precession in the 1D up conformal space.
New Geometric Methods for Computer Vision: An Application to Structure and Motion Estimation
TLDR
A coordinate-free approach to the geometry of computer vision problems is discussed, believing the present formulation to be the only one in which least-squares estimates of the motion and structure are derived simultaneously using analytic derivatives.
Quantum correlations are weaved by the spinors of the Euclidean primitives
TLDR
The resulting geometrical framework thus overcomes Bell’s theorem by producing a strictly deterministic and realistic framework that allows a locally causal understanding of all quantum correlations, without requiring either remote contextuality or backward causation.
Square Root and Logarithm of Rotors in 3D Conformal Geometric Algebra Using Polar Decomposition
TLDR
This chapter gives explicit formulas for the square root and the logarithm of rotors in 3D CGA and classifies the types of conformal transformations and their orbits.
Course notes Geometric Algebra for Computer Graphics, SIGGRAPH 2019
What is the best representation for doing euclidean geometry on computers? These notes from a SIGGRAPH 2019 short course entitled "Geometric algebra for computer graphics" introduce projective
Old Wine in New Bottles: A New Algebraic Framework for Computational Geometry
My purpose in this chapter is to introduce you to a powerful new algebraic model for Euclidean space with all sorts of applications to computer-aided geometry, robotics, computer vision and the like.
Recent Applications of Conformal Geometric Algebra
We discuss a new covariant approach to geometry, called conformal geometric algebra, concentrating particularly on applications to projective geometry and new hybrid geometries. In addition, a new
REFORM: Rotor Estimation From Object Resampling and Matching
TLDR
This paper introduces an inter-object rotor magnitude-based matching function and a subsampled iterative rotor estimation and matching algorithm that is easily parallelisable and designed for good convergence performance with models of real objects.
Transformation of Hyperbolic Escher Patterns