The Study Of Offset Curves

Offset curves,also called parallel curves,are difined as locus of points which are at constant distant d along the normal from the generator curves. Offset are widely used in various applications,such as computer graphics,3D NC machining, and so on.Therefore ,the study of offset curves has become a hot topic of CAGD.Farouki and Sakkalis introduced a class of special planar polynomial curves called Pythagorean Hodograph(PH) curves in 1990. The PH curves has polynomial parametric speed, accordingly its offset is a rational curve and its arc length is a polynomial function. Farouki also has given the sufficient and necessary condition for the polynomial parametric curves to be PH curves. PH curves has many remarkable properties,therefore it has many important applications. The cubic PH curves is the easiest curves of the PH curves,it also has many good property .it's the lowest-degree curves of all polynomial PH curves,it can approximate Bezier curves ,B-spline curves and so on.but it also has flaw,e.g.it can't deal with inflexion and self-intersection etc .The couple of planar PH cubic inherit the advantage of cubic PH curves and overcome it's weakness. It is worth to study approximatingthe Bezier curves or B-spline curves with the couple of planar PH curves .