**The Carus Mathematical Monographs**

Volume: 33;
2016;
163 pp;
Hardcover

Print ISBN: 978-0-88385-142-5

Product Code: CAR/33

List Price: $63.00

AMS Member Price: $47.25

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**Electronic ISBN: 978-1-61444-030-7
Product Code: CAR/33.E**

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# Near the Horizon: An Invitation to Geometric Optics

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*Henk W. Broer*

MAA Press: An Imprint of the American Mathematical Society

Near the Horizon starts out by considering
several optical phenomena that can occur when the sun is near the
horizon. One can sometimes see objects that are actually below the
horizon. Sometimes there seems to be a dark strip in the middle of the
solar disk. These are a result of the way that the atmosphere affects
the geometry of light rays. Broer starts his book with the Fermat
principle (rays of light take least-time paths) and deduces from it
laws for refraction and reflection; by expressing these as
conservation laws, he can handle both the case of inhomogeneous layers
of air and the case of continuous variation in the refraction index. A
surprising application is the brachistochrone problem, in which the
path of fastest descent is determined by studying how a light ray
would behave in a “flat earth” atmosphere whose refraction index is
determined by the gravitational potential. This leads to a very
interesting chapter on the cycloid and its properties.

The final chapters move from the elementary theory to a more sophisticated
version in which the Fermat Principle leads to a Riemannian metric
whose geodesics are the paths of light rays. This gives us an optics
which is geometric in a new sense, and serves as a nice demonstration
of the physical applicability of Riemannian geometry.

The book is written in a very personal and engaging style. Broer is
passionate about the subject and its history, and his passion helps
carry the reader along. The result is readable and
charming.

# Table of Contents

## Near the Horizon: An Invitation to Geometric Optics

- Cover cov11
- Half title i2
- Copyright ii3
- Title iii4
- Series iv5
- Contents vii8
- Preface xi12
- 1 Introduction and Overview 116
- I Geometry of light rays in the atmosphere 1732
- 3 Atmospheric optics 4156
- 4 Light and the cycloid 6984
- II Light rays as geodesics 97112
- 6 The brachistochrone revisited 119134
- 7 Light rays as geodesics on a surface of revolution 127142
- 8 Scholium generale 149164
- Bibliography 155170
- Index 161176