This paper presents a new method for interpolation and smooth curve fitting based on local procedures. The method is designed to produce a curve that passes through given data points and appears smooth and natural. It uses a piecewise function composed of third-degree polynomials, applicable to successive intervals of the data points. The slope of the curve at each given point is determined locally, and each polynomial representing a portion of the curve between two points is determined by the coordinates and slopes at those points. The method is based on a geometric condition for determining the slope, which is invariant under linear-scale transformations of the coordinate system. The slope of the curve at a given point is calculated using the slopes of the line segments adjacent to that point. The method is compared with other mathematical methods, and it is found to produce a curve that is closer to a manually drawn curve than those produced by other methods. The method is also applicable to multiple-valued functions, as outlined in Appendix B. The method is implemented as computer subroutines and is compared with other methods in terms of program length and computation time. The results show that the new method performs as well as or better than other methods in both interpolation and smooth curve fitting. The method is also invariant under rotation of the coordinate system and is nonlinear. The method is applicable to single-valued functions and can be implemented as computer subroutines with reasonable program length. The method is also suitable for cases where manual, but tedious, curve fitting is required. The method is described in detail in the paper, including the derivation of the slope of the curve and the interpolation between a pair of points. The method is also compared with other methods in terms of computational efficiency and accuracy. The results indicate that the new method produces a smooth, natural-looking curve that is closer to a manually drawn curve than other methods. The method is also applicable to multiple-valued functions, as outlined in Appendix B.This paper presents a new method for interpolation and smooth curve fitting based on local procedures. The method is designed to produce a curve that passes through given data points and appears smooth and natural. It uses a piecewise function composed of third-degree polynomials, applicable to successive intervals of the data points. The slope of the curve at each given point is determined locally, and each polynomial representing a portion of the curve between two points is determined by the coordinates and slopes at those points. The method is based on a geometric condition for determining the slope, which is invariant under linear-scale transformations of the coordinate system. The slope of the curve at a given point is calculated using the slopes of the line segments adjacent to that point. The method is compared with other mathematical methods, and it is found to produce a curve that is closer to a manually drawn curve than those produced by other methods. The method is also applicable to multiple-valued functions, as outlined in Appendix B. The method is implemented as computer subroutines and is compared with other methods in terms of program length and computation time. The results show that the new method performs as well as or better than other methods in both interpolation and smooth curve fitting. The method is also invariant under rotation of the coordinate system and is nonlinear. The method is applicable to single-valued functions and can be implemented as computer subroutines with reasonable program length. The method is also suitable for cases where manual, but tedious, curve fitting is required. The method is described in detail in the paper, including the derivation of the slope of the curve and the interpolation between a pair of points. The method is also compared with other methods in terms of computational efficiency and accuracy. The results indicate that the new method produces a smooth, natural-looking curve that is closer to a manually drawn curve than other methods. The method is also applicable to multiple-valued functions, as outlined in Appendix B.