Geometric Discriminate Methods of a Class of Quartic Offset-rational Curves

The necessary and sufficient conditions for quartic Bézier curves being a class of offset-rational curves are discussed.Quartic offset-rational curves which possess exact rational polynomial offsets can be classified into Pythagorean-hodograph(PH)curves,Class I ordinary OR curves and Class II ordinary OR curves,according to the decompositions of their hodograph curves in complex form.Based on the common factor of the hodograph curve and the second derivative curve,the uniform geometric discriminate methods and geometric explanations based on the control polygon for a quartic Class I ordinary OR curve or PH curve are proved.The uniform decompositions of the hodograph curve are presented and the distinguish method between Class I ordinary OR curves and PH curves is given.Finally,we show the examples of offset curves for quartic Class I ordinary OR curves and PH curves.