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The regularity is an important algebraic property for the parametric curve and surface, which depends on their parameterizations. Therefore, determining the regularity of parametric curves or surfaces is a basic problem in evaluating the quality of their parameterizations. In this paper, we present the necessary and sufficient condition for the regular Bezier curve using the Sturm's theorem. Moreover, we also give some geometric discriminant conditions, which reveal the geometric properties of the regular Bezier curve.