19 - The Rectangular Hyperbola pp. 243-252By Claudi Alsina and Roger Nelsen
Icons of Mathematics
Dolciani Mathematical Expositions
Online ISBN: 9780883859865
Chapter DOI: http://dx.doi.org/10.5948/UPO9780883859865.021
Subjects: Geometry and topology
What mathematician has ever pondered over an hyperbola, mangling the unfortunate curve with lines of intersection here and there, in his efforts to prove some property …?Lewis CarrollThe Dynamics of a Parti-cle(1865)
Hyperbolas, along with circles, ellipses, and parabolas, belong to the family of plane curves known as conic sections. Menaechmus, Euclid, and Aristaeus studied conic sections, and Apollonius of Perga (circa 240-190 BCE) wrote Conics, an eight-volume work that presented the modern form of characterizing these curves as intersections of cones with planes.
Hyperbolas appear in nature and in man-made objects. From left to right in Figure 19.1 we see the hyperbolic path taken by some single-apparition comets, three of the six hyperbolic arcs at the tip of a sharpened hexagonal pencil, and the hyperbolic shape of the light beam of a flashlight held parallel to a wall.
Striking examples of the use of the hyperbola in architecture appear in the design of the columns of the Catedral Metropolitana Nossa Senhora Aparecida in Brasília, and in the profile of the McDonnell Planetarium in St. Louis, as illustrated in Figure 19.2.
A rectangular hyperbola (or equilateral or right hyperbola) is one whose asymptotes are perpendicular.We will explore the role played by rectangular hyperbolas in the mathematics of logarithms, inequalities, and hyperbolic functions.
Hyperbolas and bells
The first book addressing the problem of finding the ideal shape for a bell was Harmonicorum, libri XII (1648), a beautiful study by Marin Mersenne (1588–1648) searching for the functional description of the profile of a bell.
Twenty Key Icons of Mathematics:
The Bride's Chair:
Zhou Bi Suan Jing:
The Right Triangle:
Arcs and Angles:
Polygons with Circles:
Yin and Yang:
The Rectangular Hyperbola:
Solutions to the Challenges:
About the Authors: