Geometric interpolation by PH curves with quadratic or quartic rational normals

Pythagorean-hodograph (PH) curves have nice properties and have found important applications in geometric modeling and CNC machining. While the unit normals of PH curves of degree n are generally rational curves of degree n−1, this paper investigates PH curves of arbitrary degrees but with only quadratic rational unit normals when the curves are convex or with quartic rational unit normals when the curves have single inflection points. PH curves with quadratic or quartic rational normals have simple Gauss maps and hodographs of the curves are given by low degree tangent vector fields together with simple real scaling functions. Practical algorithms for interpolation of point-normal pairs or point-normal-curvature pairs together with unit normals at selected parameter coordinates or at inflection points by the investigated PH curves without or with the constraint of arc lengths have been given. The parameters for defining the interpolating PH curves are either obtained directly from the input data or by solving simple linear systems. This method of PH curve interpolation has unique solutions and the shapes of the interpolating PH curves are controlled well by the interpolated data. Even though the interpolating curves may have cusps, the regularity of the PH curves can be checked easily based on the signs of the real scaling functions within the hodographs.