# Beyond the ABCDs: A projective geometry treatment of paraxial ray tracing using homogeneous coordinates

@inproceedings{Corcovilos2022BeyondTA, title={Beyond the ABCDs: A projective geometry treatment of paraxial ray tracing using homogeneous coordinates}, author={Theodore A. Corcovilos}, year={2022} }

Homogeneous coordinates are a projective geometry tool particularly well suited to paraxial geometric optics. They are useful because they allow the expression of rotations, translations, affine transformations, and projective transformations as linear operators (matrices). While these techniques are common in the computer graphics community, they are not well-known to physicists. Here we apply them to paraxial ray tracing. Geometric optics is often implemented by tracing the paths of non…

## References

SHOWING 1-10 OF 30 REFERENCES

3D Oriented Projective Geometry Through Versors of R

- Mathematics
- 2015

It is possible to set up a correspondence between 3D space and R , interpretable as the space of oriented lines (and screws), such that special projective collineations of the 3D space become…

Projective Formulation of the Problems of Geometrical Optics. I. Theoretical Foundations

- Physics
- 1959

A procedure is described and discussed, by which the requirements imposed on an optical system can be formulated in terms of the unknown surfaces, so that conditional (differential) equations can be…

Poisson commutator-anticommutator brackets for ray tracing and longitudinal imaging via geometric algebra

- Physics
- 2008

We use the vector wedge product in geomet- ric algebra to show that Poisson commutator brackets measure preservation of phase space areas. We also use the vector dot product to define the Poisson…

Representation of Gaussian beams by complex rays.

- PhysicsApplied optics
- 1985

This paper shows that a fundamental Gaussian beam propagating in a lenslike medium with cylindrical symmetry can be generated by the rotation about its axis of a skew ray which obeys the laws of…

6×6 matrix formalism of optical elements for modeling and analyzing 3D optical systems

- Physics
- 2009

The 6×6 matrix formalism is utilized for optical boundary surfaces to develop a new approach for modeling and analyzing 3D optical systems comprising multiple lenses and/or mirrors and clarifying many of the system’s basic properties.

Paraxial meridional ray tracing equations from the unified reflection-refraction law via geometric algebra

- Physics
- 2008

We derive the paraxial meridional ray tracing equations from the unified reflection-refraction law using geometric algebra. This unified law states that the nor- mal vector to the interface is a…

Lens-System Diffraction Integral Written in Terms of Matrix Optics

- Physics
- 1970

This paper deals with the propagation of coherent light through lens systems; it relates diffraction theory to ray optics. A diffraction integral is derived which relates the electromagnetic fields…

A geometric algebra reformulation of geometric optics

- Mathematics, Physics
- 2004

We present a tutorial on the Clifford (geometric) algebra Cl3,0 and use it to reformulate the laws of geometric optics. This algebra is essentially a Pauli algebra, with the Pauli sigma matrices…

Minimal optical decomposition of ray transfer matrices.

- PhysicsApplied optics
- 2008

It is found that general lossless one-dimensional optical systems can be synthesized with a maximum of four elements and two-dimensional Optical Systems can be synthesisized with six elements at most.