# Quantitative Considerations

• Arlan Ramsay
• Robert D. Richtmyer
Part of the Universitext book series (UTX)

## Abstract

By use of material in the two preceding chapters, quantitative formulas are derived for many of the objects and relations described qualitatively in Chapter 3. First, a formula is given for the angle of parallelism, which tells in what direction a line m has to be started from a point A to be asymptotic to another line ; it is expressed as a function of the distance y from A to , obtained by dropping a perpendicular from A to . From that, an equation in polar coordinates is derived for a horocycle, which is in a sense the limit of a circle as its radius goes to infinity. Next, differential equations and formulas (the latter obtained by solving the differential equations) are obtained for the functions g(r) and f (r) in terms of which the length of a circular arc of radius r and the area of a circular sector of radius r were expressed in Chapter 3. Then, formulas are derived for the legs a, b of a right triangle (not assumed small) of hypotenuse c and one acute angle. The formulas contain the hyperbolic functions sinh and tanh of a, b, and c. A generalization of the law of cosines is found, which gives the length of the third side of a general triangle in terms of two sides a, b and included angle. A formula for a line in polar coordinates is given, and the equation for an equidistant. An equidistant is a curve at a fixed distance from a given line (obtained by dropping perpendiculars); an equidistant is not itself a (straight) line. Ideal points at infinity are defined, and the chapter closes with formulas for certain isometries (especially translations) in polar coordinates.

## Keywords

Ideal Point Hyperbolic Plane Isosceles Triangle Circular Sector Small Triangle
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer Science+Business Media New York 1995

## Authors and Affiliations

• Arlan Ramsay
• 1
• Robert D. Richtmyer
• 1
1. 1.Department of MathematicsUniversity of ColoradoBoulderUSA