| In computer aided design systems,the planner OR curves are a kind of parametric curves with rational offset curves.They can be divided into two types:the first type is the PH curves,whose offset curves are rational.The second type is the indirect-PH curves,whose offset curves are rational after an appropriate reparameterization.The Bézier curves are widely used in computer aided design systems.However,the offset curves of a planar Bezier curve are not necessarily rational polynomials in general.We studies a class of cubic indirect-PH curve.Firstly,based the geometric properties of cubic indirect-PH curves,we investigate the a geometric characterization of this curve,and propose a sufficient and necessary condition for a cubic plane Bezier curve to be an indirect-PH curve.Secondly,we study the sufficient conditions of monotone curvature of such curves by calculating the curvature formula of cubic indirect-PH curve.On the one hand,the freedom of parameters required for this condition only involves solving cubic equations;On the other hand,this condition can not only be used to judge the curvature monotonicity of cubic indirect-PH curves,but also be used to construct a cubic indirect-PH curve with monotonous curvature satisfying G1 Hermite interpolation requirement.Finally,we use the cubic indirect-PH curve with monotonous curvature to construct G1 Hermite interpolation and C-type transition curve between two circles with given radius of curvatures.We show three experiments for the transition curve construction,and compare the results with the existing work.The construction of transition curves does not resort to solve any equations,which greatly reduces the complexity of the calculations.Furthermore,our method is able to use lower degree of curves to construct transition curves than previous method. |
