Near the Horizon: An Invitation to Geometric Optics

This new book in the MAA’s Carus Mathematical Monograph series uses the investigation of the propagation of the sun’s light near the horizon as the inspiration for studying light rays as geodesics. Geometric optics is well suited for explaining phenomena that include refraction and reflection — including handling curved paths in regions where the refractive index changes — but it does not incorporate optical effects arising from diffraction or interference. It is, however, particularly valuable for studying optical aberrations.

The author found inspiration for his book in Marcel Minneart’s work, perhaps best represented by the beautiful book Light and Color in the Outdoors. Minnaert presents photographs, drawings and descriptions of many fascinating optical effects observed outdoors, many of which involve the sun. The author of the current book takes a part of this — the behavior of light paths near the horizon — and presents a mathematical treatment that is accessible to undergraduates. Part I of the book considers the geometry of light rays in the atmosphere; it uses basic calculus and some Euclidean geometry. The analysis is deeper in Part II where the discussion moves to consideration of curved light paths using variational methods and differential geometry.

The main thrust of Part I is to understand near-horizon optical phenomena that include deformations of the image of the sun’s disk just before sunset as well as associated mirages. To do this, the author develops and applies the tools of geometric optics to study how light rays are refracted and reflected through layers of atmosphere. The relationship between temperature and refractive index is critical. Usually temperature decreases monotonically with height above the earth, but temperature inversions can create a discontinuity in the refractive index profile. Consequences of this include a strip missing from the image of the setting sun and the sun appearing to set before it reaches the horizon.

The geometric setting for Part I is a vertical plane that includes the eye of an observer and the center of the object (e.g., the sun) being observed. The author begins with a flat earth model where the atmosphere is divided into several distinct horizontal layers each having its own index of refraction. He proceeds similarly with a round earth model where the layers are concentric and then lets the refractive index vary continuously through the atmosphere. Using Fermat’s Principle he derives equations for the path of light through the atmosphere.

Most optical phenomena near the horizon can be explained using this approach. The famous “green flash” occurs when the red component of sunlight is refracted more strongly than the green, so that under certain special conditions only the green of the setting sun is visible.

Variational principles are the focus of Part II. Earlier in Part I the author discusses Bernoulli’s brachistochrone problem and describes how a certain refractive index profile for the flat earth model could create a cycloid-like light path. In Part II he continues to explore the consequences of viewing light paths as geodesics. He moves from Fermat’s Principle to the Euler-Lagrange equations and then to Hamilton’s variational principle as he explores geometric optics via methods like those used Hamiltonian mechanics. By the end of the book he is considering whether the geometric optics of the round earth atmosphere could be considered a geodesic problem on a surface of revolution and whether it could be isometrically embedded therein.

I found the book somewhat disappointing. So much of the book is devoted to variational principles and their consequences that optics — nominally the primary subject — gets short shrift. Part I was promising but could have provided more details on the optical consequences of the models.

The many digressions also have the effect of obscuring the author’s primary exposition. While the isometric embedding theorem is interesting in its own right, for example, it isn’t at all clear why it’s particularly relevant to the optics.

Bill Satzer (bsatzer@gmail.com) was a senior intellectual property scientist at 3M Company. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.