In this paper, we develop a simple equation for inflection points and singular pointsof the planar quartic Bezier curve and discuss their distribution. We suggest a clas-sification for the planar quartic Bezier curve by the number of inflection points andgive a sufficient and necessary condition for the existence of singular points (i.e. knotsand cusps). With the numerical examples we show that the convex Bezier curve canbe produced not only from the nonconvex simple characteristic polygon but also fromthe non-simple characteristic polygon and that singular points (knots) and inflectionpoints may coexist for the planar quartic Bezier curve, although this case will neveroccur for the planar cubie Bezier curve.