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.
KeywordsIdeal Point Hyperbolic Plane Isosceles Triangle Circular Sector Small Triangle
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