Research On The Problem Of The Hermite Interpolation By PH 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. Farouki and Sakkais 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. PH curves has many remarkable properties, therefore it has many important applications.This paper using complex analysis and curve integration, the construction of PH curve of rational cubic, quintic and sevenfold which satisfied Hermite interpolation conditions are studied in this paper and their corresponding Bezier representation are derived. The PH curve of quintic and sevenfold have continuous unit tangents and signed curvature, and their arclength functions are the polynomial of their parameters. The PH curve of quintic and sevenfold have offset curve that admit exact rational algebraic representation, intuitive geometrical interpretation and can flexibly deal with inflection point. For the PH curve of quintic with eight degrees of freedom, and often have the similar nature of curve of cubic parametric, but eliminates in its two vertices only has the first-order continual derivative. But the PH curve of sevenfold have 10 degree of freedom, and has the second-order continual derivative in the vertex place, it has the obvious superiority in the reality production of the life. For instance, the PH curve of sevenfold has application in railroad construction.