Introduction to Hyperbolic Geometry pp 190-201 | Cite as

# Quantitative Considerations

## 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 *ℓ*; it is expressed as a function of the distance *y* from *A* to *ℓ*, obtained by dropping a perpendicular from *A* to *ℓ*. 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## Preview

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