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eBook ISBN:  9781614440307 
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AMS Member Price:  $52.50 
Hardcover ISBN:  9780883851425 
eBook: ISBN:  9781614440307 
Product Code:  CAR/33.B 
List Price:  $145.00 $110.00 
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AMS Member Price:  $108.75 $82.50 
Hardcover ISBN:  9780883851425 
Product Code:  CAR/33 
List Price:  $75.00 
MAA Member Price:  $56.25 
AMS Member Price:  $56.25 
eBook ISBN:  9781614440307 
Product Code:  CAR/33.E 
List Price:  $70.00 
MAA Member Price:  $52.50 
AMS Member Price:  $52.50 
Hardcover ISBN:  9780883851425 
eBook ISBN:  9781614440307 
Product Code:  CAR/33.B 
List Price:  $145.00 $110.00 
MAA Member Price:  $108.75 $82.50 
AMS Member Price:  $108.75 $82.50 

Book DetailsThe Carus Mathematical MonographsVolume: 33; 2016; 163 pp
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 leasttime 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

Chapters

Chapter 1. Introduction and Overview

I. Geometry of light rays in the atmosphere

Chapter 2. Geometric optics

Chapter 3. Atmospheric optics

Chapter 4. Light and the cycloid

II. Light rays as geodesics

Chapter 5. The principles of Fermat and Hamilton

Chapter 6. The brachistochrone revisited

Chapter 7. Light rays as geodesics on a surface of revolution

Chapter 8. Scholium generale


Additional Material

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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 leasttime 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.

Chapters

Chapter 1. Introduction and Overview

I. Geometry of light rays in the atmosphere

Chapter 2. Geometric optics

Chapter 3. Atmospheric optics

Chapter 4. Light and the cycloid

II. Light rays as geodesics

Chapter 5. The principles of Fermat and Hamilton

Chapter 6. The brachistochrone revisited

Chapter 7. Light rays as geodesics on a surface of revolution

Chapter 8. Scholium generale