Abstract: The structure of the common mapping curve is established and analyzed. The mapping theorem can be obtained plainly by starting from the concept of the loop-line,followed by linking with the zeroes and poles of complex function. Based on the numeral and interactive relationship followed by zeroes, poles and the loop-line, the interactive conditions of the formation of the mapping theorem is determined, by which the uncertainty in understanding the conclusion of N=Z-P can be avoided. In the case of the clear existence of zeroes shown by the mapping curve, the procedures of judging whether the "Z" is existent or not by N=-P can also be saved. When the number of curve anti-clockwise loops is more than given poles, the title itself should be wrong According to the restricted conditions.