This applet represents the Poincaré model of the hyperbolic plane, which corresponds to the white interior of the pictured circle.

You can explore many aspects of hyperbolic geometry, e.g.:[br][list][br][*] examine the sum of the interior angles of triangles observing, in particular, what happens when the sides of the triangle become very small;[br][*] given a point [math]P[/math] exterior to a line [math]r[/math], construct the perpendicular [math]p[/math] to [math]r[/math] passing through [math]P[/math], and then the perpendicular [math]\pi[/math] to [math]p[/math] passing through [math]P[/math] (this is the Euclidean construction of a parallel);[br][*] given a point [math]P[/math] exterior to a line [math]r[/math], construct as before the perpendiculars [math]p[/math] and [math]\pi[/math], select any point [math]Q[/math] on [math]r[/math], draw the perpendicular [math]s[/math] to [math]\pi[/math] passing through [math]Q[/math], and the circle [math]\gamma[/math] with center [math]P[/math] and radius [math]HQ[/math], where [math]H[/math] is the intersection of [math]r[/math] and [math]p[/math]. Name [math]B[/math] the point of intersection of [math]\gamma[/math] and [math]s[/math] which lies between [math]r[/math] and [math]\pi[/math], and draw the line [math]b[/math] passing through [math]P[/math] and [math]B[/math]. What can you say about [math]b[/math]?[br][/list][br][br]How many lines exist that are parallel to a given line [math]r[/math] and pass through an exterior point [math]P[/math]?[br]The latter is the János Bolyai construction of the [i]asymptotic parallel line[/i] (there are two such lines for any [math]r[/math] and [math]P\not\in r[/math].