The Center for Undergraduate Research in Mathematics (aka CURM) funded two research projects that my colleague, Dr. Emek Kose, and I conducted with two different groups of St. Mary’s students.  These projects focus on an exciting and novel area of mathematics that lies at the intersection of both differential geometry and optics.

We are interested in designing curved mirror surfaces that re-direct light ways towards a camera — such a camera-mirror pair is a basic example of a  catadioptric sensor.  In particular, we want to design these surfaces so that the camera can produce images with very large fields of view but features little to no distortion.  As anyone who has ever gazed into a fun-house mirror can attest, though, some kinds or types of distortion are necessary when using curved surfaces.  For instance, a mirror may result in images wherein angles are distorted, lengths are distorted, and/or areas are distorted.  While it is often possible to eliminate a single kind of distortion, doing so typically comes at the cost of increasing another.

Emek Kose, Caroline Van Blargan, Dan Carroll, Nora Stack, and Me

St. Mary’s students Caroline VanBlargen (2014), Nora Stack (2014), and Dan Carroll (2014) worked as members of our 2013-2014 CURM group, and, in addition to presenting findings at various conferences, this project resulted in the publication Equitable Mirrors in the journal Applied Optics.  This article explores some of the finer properties of so-called “equi-areal mirrors” — those that uniformly rescale areas — and introduced techniques for developing mirror surfaces that simultaneously minimize angular and areal distortion.  Indeed, the image atop this very web page was taken from this paper; it is obtained by using one of the new mirrors we designed as part of a catadioptric sensor that photographs a spherical room.  The nearly-uniform spacing of concentric circles indicates that this mirror very nearly preserves angles.

St. Mary’s students Jared Saltzberg (2017) and Anna Steinfeld (2017) worked as members of our 2014-2015 CURM group, and they designed mirrors that, when used as part of a catadioptric sensor, can image globe-painted spheres in the same ways that cartographers created flat maps of the Earth.  The results of their work have also been presented at various conferences and have been collected into a manuscript that is currently under review.  These findings establish exciting connections between famous geometric surfaces and famous maps of the earth; for example, Mercator’s map can be obtained by using the graph of $f(x,y) = e^x\cos y = \text{Re}(e^z)$.  A perhaps more surprising example is that the Azimuthal equidistant map can be obtained by using a rotationally symmetric mirror whose profile curve goes by the name “the grim reaper” (it  translates under mean curvature flow).  The results of this mirror are shown in the image below.

Mirror surface (left) that induces the Azimuthal Equidistant projection (right)

There are a number of ways in which to continue this research, and I remain excited to pursue any or all of them.  How might the design of these mirrors change if light rays are allowed to bend, for example?  How might one best discretely approximate such mirrors?

I am currently working on some of these questions and will update this page as results are available.

— Casey