Abstract: The Pythagorean Bézier hodograph curves are Bézier curves ｛x（t） ，y（t）｝，whose hodograph （first-order parametric derivative） components satisfy the Pythagorean condition x²（t） ＋y²（t） ＝σ²（t） for some polynomial σ（t） ． For PB curve with degree n（n is odd） ，its offset curve is represented by rational Bézier curve with （2n-1）-th degree and the arc length is represented by polynomial．Especially，an explicit representation of quintic PB curves is established and the sufficient and necessary algebraic characterization of quintic PB curves is presented．The characterization of its shape and the condition of inflections is discussed．Their offset curves exactly represented by rational Bézier curves of nine-degree and the arc length represented by the polynomial form are also obtained．