A piecewise algebraic curve is de1ned by a bivariate spline function.Using the techniques of the B-net form of bivariate splines function, discriminant sequence of polynomial (cf.Yang Lu et al.(Sci.China Ser.E 39(6) (1996) 628) and Yang Lu et al.(Nonlinear Algebraic Equation System and Automated Theorem Proving, Shanghai Scienti1c and Technological Education Publishing House, Shanghai, 1996)) and the number of sign changes in the sequence of coeffecients of the highest degree terms of sturm sequence, we determine the number of real intersection points of two piecewise algebraic curves whose common points are 1nite.A lower bound of the number of real intersection points is given in terms of the method of rotation degree of vector field.