Abstract: To develop the geometric properties of the Bézier-like systems for non-polynomial spaces,researches are done for p-Bézier curves of the linear trigonometric polynomial space.Firstly,a parameter equation of an ellipse is changed into the p-Bézier form from geometric transformations and parameter transformations.Secondly,according to comparison,a conclusion that every linear p-Bézier curve except degenerated cases is an elliptic arc is drawn.Thirdly,the relations between the geometric elements of the ellipse,such as the center,foci,vertices of major and minor axes,and the control points are given.Lastly,a sufficient and necessary condition under which a linear p-Bézier curve is an arc is given.Moreover,example analysis shows that these geometric elements can be represented as linear interpolation forms of control points.