Spatial PH Curves Based On Quaternion And Bézier Method

Pythagorean-hodograph(PH)curve is a polynomial parametric curve,which has the advantage that arc length and offset curves can be accurately represented.The unique construction method of PH curve has many good properties,which make PH curve has important application value in Computer Aided Geometric Design.The algebraic structure of spatial PH curve is more compact by quaternion,and the Bézier method is more suitable for shape design.On this basis,the spatial quintic and septic PH curves are constructed according to the given spatial vertex sequence.Based on the polynomial quaternion,the Bernstein basis function is used as the underlying framework to construct the Bézier curves,which are integrated to obtain the spatial PH curves represented by control vertices.The curve control vertex adds two new nodes on the basis of the given spatial vertex sequence and describes the relationship between the vertices by quaternion multiplication.In the process of constructing spatial PH curve,several free parameters are introduced,and the spatial PH curve can be adjusted and optimized by the constraint and reasonable selection of these free parameters.The free parameters introduced by quaternion orthogonal rotation are constrained from the point of view of reducing curve flexure and ensuring curve smoothness.In addition,the free parameters of the curve can be reasonably selected by arc length of curve or convex hull property of Bernstein basis function.Finally,ER marker is selected for spatial PH curve,and the angular velocity component of the frame vector is used to describe the rotation of the PH curve in space,so as to achieve the minimum rotation smoothing of spatial PH curve.In the experiment,the optimized spatial PH curve and its Bézier control polygon meet the requirements of smoothness of the curve and reduce the torsion of PH curve in space.Moreover,the spatial PH curve constructed by this method is consistent with that of the Bézier method,the geometric meaning is clear,the method is effective and easy to implement.