### Abstract

We present a tutorial on the Clifford (geometric) algebra Cl_{3,0} and use it to reformulate the laws of geometric optics. This algebra is essentially a Pauli algebra, with the Pauli sigma matrices interpreted as unit rays or vectors. In this algebra, the exponentials of imaginary vectors act as vector rotation operators. This property lets us rewrite the laws of reflection and refraction of light in geometric optics in exponential form. The reformulated laws allow easy translation of symbols to words and to diagrams. They also are shown to be equivalent to standard vector formulations. These coordinate-free laws can be shown to simplify the analysis of geometric optics problems such as the tracing of meridional and skew rays in lenses and optical fibers.

Original language | English |
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Pages (from-to) | 92-97 |

Number of pages | 6 |

Journal | American Journal of Physics |

Volume | 72 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 1 2004 |

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### All Science Journal Classification (ASJC) codes

- Physics and Astronomy(all)

### Cite this

*American Journal of Physics*,

*72*(1), 92-97. https://doi.org/10.1119/1.1621029

**A geometric algebra reformulation of geometric optics.** / Sugon, Quirino Jr Mallorca; McNamara, Daniel J.

Research output: Contribution to journal › Article

*American Journal of Physics*, vol. 72, no. 1, pp. 92-97. https://doi.org/10.1119/1.1621029

}

TY - JOUR

T1 - A geometric algebra reformulation of geometric optics

AU - Sugon, Quirino Jr Mallorca

AU - McNamara, Daniel J.

PY - 2004/1/1

Y1 - 2004/1/1

N2 - 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 interpreted as unit rays or vectors. In this algebra, the exponentials of imaginary vectors act as vector rotation operators. This property lets us rewrite the laws of reflection and refraction of light in geometric optics in exponential form. The reformulated laws allow easy translation of symbols to words and to diagrams. They also are shown to be equivalent to standard vector formulations. These coordinate-free laws can be shown to simplify the analysis of geometric optics problems such as the tracing of meridional and skew rays in lenses and optical fibers.

AB - 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 interpreted as unit rays or vectors. In this algebra, the exponentials of imaginary vectors act as vector rotation operators. This property lets us rewrite the laws of reflection and refraction of light in geometric optics in exponential form. The reformulated laws allow easy translation of symbols to words and to diagrams. They also are shown to be equivalent to standard vector formulations. These coordinate-free laws can be shown to simplify the analysis of geometric optics problems such as the tracing of meridional and skew rays in lenses and optical fibers.

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U2 - 10.1119/1.1621029

DO - 10.1119/1.1621029

M3 - Article

AN - SCOPUS:0942278794

VL - 72

SP - 92

EP - 97

JO - American Journal of Physics

JF - American Journal of Physics

SN - 0002-9505

IS - 1

ER -