Pedal Curves and their Envelopes

In the Euclidean plane with the pedal transformation the pedal curve of a given generating curve respect to the pole can be obtained. The pedal curve ce of a generating curve c1 with respect to a pole P is the locus of the foot of the perpendicular lines from P to all tangent lines of the curve c1, [4]. If the generating curve c1 is a conic then its pedal curve ce can be given as an envelope of circles, [2]. The pedal transformation can be extended in the quasi-hyperbolic plane where the metric is induced by the absolute gure FQH = fF ; f1 ; f2g. In the quasi-hyperbolic plane the pedal curve cqh of a given generating curve c2 respect to the polar line p is the locus of the lines joining the points of the curve c2 with its corresponding perpendicular points on the polar line p, [1]. In this presentation we will give the construction of the envelope of the pedal curve and study the pedal curve as an envelope of circles in the quasi-hyperbolic plane.