Frobenius Integrability, Automotive Blind Spots, Non-reversing Mirrors, and Panoramic Mirrors

Abstract

When an observer looks at a curved mirror, they may sense that a nonlinear map is at work. Here we consider the problem of finding the mirror that realizes a given map. The natural language for such problems is that of planar distributions, and one tool for testing for the existence of solutions is the Frobenius theorem. For situations where exact solutions do not exist, we describe an approximation method that can give good results for applications. Our examples will include non-reversing mirrors, panoramic mirrors, and automotive mirrors without blind spots.

MSC:

• 78A05

Acknowledgment

The second author wishes to thank Ze-Li Dou, Bill Plummer, Raymond Thomas, and Hiroshi Tsuda.

Notes

1 A thorny terminology issue crops up here. A distortion map can, for example, be a scaling, which doesn’t sound very distorting. To avoid confusion, we reserve the word “distortion” for use in the phrase “distortion map,” and never use it informally. We may use terms like “fisheye-like” to indicate that a map is not a perspective projection. Saying that a map is “not distorting” could mean many things, e.g., area preserving, conformal, etc. See [11].

2 While we work with vector fields, for multiple mirror problems the viewpoint of Élie Cartan, i.e., forms and exterior differential systems, is indispensable [20, 19, 8].

3 We are not telling the whole story. If you assume $\left|\mathbf{W}\right|=1$ then you can find a best γ.

4 Although they do not all give rise to integrable distributions.

Additional information

Notes on contributors

Elim Hicks

ELIM HICKS is an undergraduate at the Cooper Union, where he is studying engineering.

The Cooper Union, 30 Cooper Sq, New York, NY 10003

[email protected]

R. Andrew Hicks

R. ANDREW HICKS received his undergraduate degree from Queens College CUNY in 1988 and his doctorate from the University of Pennsylvania in 1995. He spent three years as a postdoc in the UPenn GRASP laboratory, working on vision based control of robots, which led him to the subject of optical design. He has been at Drexel since 1999.

Drexel University, Philadelphia PA 19104

[email protected]

Ron Perline

RON PERLINE received his undergraduate degree from UC Santa Cruz, and his Ph.D. from UC Berkeley. He returned to Santa Cruz as a Visiting Assistant Professor, and then joined the math faculty at Drexel University. He is interested in applied mathematics in various contexts, including integrable systems and differential geometry, and (of course) mirror mathematics.

Drexel University, Philadelphia PA 19104

[email protected]

Sarah G. Rody

SARAH G. RODY studied math, music, and conflict studies at Goshen College before receiving her Ph.D. in mathematics from Drexel University. She now teaches at Chestnut Hill College.

Chestnut Hill College, Philadelphia PA 19104

[email protected]