Research On The Geometric Characteristic Of Indirect-PH Curves And Related Problems

Offsetting has many applications in CAD such as motion planning,CNC machining,railway design etc.However,it is known that a planner polynomial curve in general may not have a rational offset.The curves that have rationally parameterized offsets can be classified into two classes: PH curves and indirect-PH curves.The offset curve of PH curve is rational and the offset curve of indirect PH curve is rational after an appropriate parameterization.The algebraic characteristics of planar polynomial curves with rational offsets cannot be simply and intuitively translated into geometric constraints corresponding to control polygons.Based on the complex representation of curves,the geometric characteristics of indirect PH curves are studied in this thesis.A more succinct proof by introducing auxiliary control points is provided in this thesis.Specific research contents are as follows:1.The cubic indirect PH curve is studied by introducing auxiliary control points.The algebraic structure of indirect PH curve is transformed into geometric characteristic conditions,which are represented by the control polygon.By introducing the real parameter,we get more concise and equivalent sufficient and necessary conditions of the geometric characteristic.The real parameters can be used as shape adjusting factors to adjust the shape of the curve.2.Based on the geometric characteristic conditions of cubic indirect PH curves,a new construction algorithm is proposed.By introducing real parameters and selecting different shape adjusting factors,the cubic indirect PH curves of different shapes can be constructed.3.The geometric characteristics of quartic indirect PH curves are represented.Quartic indirect PH curves can be divided into two types according to the derivatives.Based on the algebraic conditions of indirect PH curves,Bernstein polynomial and derivative properties of indirect PH curve are used.By introducing auxiliary control points,algebraic characteristics of quintic PH curves can be transformed into nonlinear complex systems with respect to control points.The conditions for a quartic planar curve to be an indirect PH curve can be described in terms of geometric constraints on its control polygon.