the ideal (end) points of the line. the point where these tangents intersect is the pole. For diameters of the disk: the pole is at infinity perpendicular to the diameter. To construct a perpendicular to a given line through a given point draw the ray from the pole of the line through the given point. The part of the ray that is inside the disk is the perpendicular. When the line is a diameter of the disk then the perpendicular is the chord that is (Euclidean) perpendicular to that diameter and going through the given point. To find the midpoint of given segment : Draw the lines through A and B that are perpendicular to . (see above) Draw the lines connecting the ideal points of these lines, two of these lines will intersect the segment and will do this at the same point. This point is the (hyperbolic) midpoint of. To bisect a given angle : Draw the rays AB and AC. Draw tangents to the circle where the rays intersect the boundary circle. Draw a line from A to the point where the tangents intersect. The part of this line between A and the boundary circle is the bisector. The common perpendicular of two lines is the chord that when extended goes through both poles of the chords. When one of the chords is a diameter of the boundary circle then the common perpendicular is the chord that is perpendicular to the diameter and that when lengthened goes through the pole of the other chord. To reflect a point P in a line l: From a point R on the line l draw the ray through P. Let X be the ideal point where the ray intersects the absolute. Draw the ray from the pole of line l through X, let Y be another ideal point that intersects the ray. Draw the segment RY. The reflection of point P is the point where the ray from the pole of line l through P intersects RY. Circles, hypercycles and horocycles While lines in the hyperbolic plane are easy to draw in the Klein disk model, it is not the same with circles, hypercycles and horocycles. Circles (the set of all points in a plane that are at a given distance from a given point, its center) in the model become ellipses increasingly flattened as they are nearer to the