Apollonius meets Pythagoras1 Ching-Shoei Chiang* and Christoph M. Hoffmann† * Soochow University, Taipei, Taipei, R.O.C. E-mail:
[email protected] Tel/Fax: +886-2-2311-1531 ext. 3801 † Purdue University, West Lafayette, IN 47906, USA E-mail:
[email protected] Tel: 1-765-494-6185 Abstract—Cyclographic maps are an important tool in rational Bézier curve. Section 4 illustrates the Apollonius Laguerre geometry that can be used to solve the problem of problem of 3 primitives, including ray, cycle, and oriented PH Apollonius. The cyclographic map of a planar boundary curve is curve. We conclude with results in the final section. a ruled surface that we call -map. A PH curve is a free form curve whose tangent and normal are polynomial. This implies that its cyclographic map has a parametric form. This paper considers the medial axis transform (MAT) of a PH curve with a II. DEFINITIONS AND THEOREMS ray and the MAT of a PH curve with a cycle. We transform the Pythagorean Hodograph(PH) has a property that is closely MAT finding problem into a surface intersection problem. Here, linked to the its cyclographic map. If a 2D curve has the PH the intersection curve is representable in rational Bezier form. property, that is, if its tangent is a polynomial, then the Furthermore, the generalized Apollonius problem concerning rays, cycles, and oriented PH curves is introduced. cyclographic map for the curve has a polynomial form. We define some key terms now, including the Pythagorean Hodograph curve and the cyclographic map. I. INTRODUCTION A.