Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1994. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 22)

SZILASSI, L., A computer-aided.démonstration of the Poincare model of hyperbolic geometry

138 The loci of both points with the same abscissa (Une) and points with the same ordinate (hypercycle) will be an arc < representing the line or arc h representing the hypercycle, respectively, on the P-model. The intersection of these arcs will be the point corresponding to (•' .//), i.e. the coordinates {.r

k. y

k) in the Cartesian systein with origin O, which we had to determine from the parameters <"/(•' ), d(y) and r. Using the notation of the figure we get the équations of circles > and fi: (x — Cf. y -i- y

2 = p

2 where c

t = p, -t- <l(x) — () K,

, x

2 -f {y + Ch)

2 = pi where c

h = p

h - d(y) = //K

h. It also holds that c

2 = p

2 + T

2 because ,s is perpendicular to the circle of inversion k, and c\ — p\ — r

2 as h intersects k in opposite points. From the above we have c

( x -f Cf,y = r

2 for the radical axis of circles < and h. Also, as e ± h

, the points O and P both he on the Thaïes' circle of the segment A'gA'/,, so the angles OK

eP < and O K h P < are equal. Hence: _ ph p, ' Calculation shows that _ r

2-d(x)-(r

2+d

2(y)

) _ r

2 • d(y) • ((r

2 + d

2(x)) X

k r

4 + d.

2(x) • d

2(y) V

k r* + d

2{x) • d*(y) ' As both d(x) and d(y) contain r as coefficient, these formuláé can be simplifie d: th{q-x) • (1 + th

2(q -y)

) _ th(q • y) • (l + th

i(q>x)) X

k~ l + th

2{q-x)-th

2{q-y)

V' V

k ~ 1 + th

2 {q • x) • th

2 {q • y) "