HIROSHI AKIMA presents a new mathematical method for interpolation and smooth curve fitting from a given set of data points in a plane. The method is designed to ensure that the resulting curve passes through all the given points and appears smooth and natural. It is based on a piecewise function composed of third-degree polynomials, with the slope of the curve determined locally at each given point. The method is compared with other mathematical methods, and it is shown that the curve obtained by this new method is closer to a manually drawn curve. The paper also discusses the computer implementation of the method, including program lengths and computation times, and provides a detailed comparison with other methods. The method is applicable to both single-valued and multiple-valued functions, with specific modifications outlined for the latter. The author concludes by highlighting the advantages of the new method, such as its smoothness, natural appearance, and computational efficiency.HIROSHI AKIMA presents a new mathematical method for interpolation and smooth curve fitting from a given set of data points in a plane. The method is designed to ensure that the resulting curve passes through all the given points and appears smooth and natural. It is based on a piecewise function composed of third-degree polynomials, with the slope of the curve determined locally at each given point. The method is compared with other mathematical methods, and it is shown that the curve obtained by this new method is closer to a manually drawn curve. The paper also discusses the computer implementation of the method, including program lengths and computation times, and provides a detailed comparison with other methods. The method is applicable to both single-valued and multiple-valued functions, with specific modifications outlined for the latter. The author concludes by highlighting the advantages of the new method, such as its smoothness, natural appearance, and computational efficiency.