Step 1 - Data and Initialization:

You have a set of data points (_{x0, y0}), (x₁, y₁), …, (x_n, y_n) that you want to interpolate. These points represent the values of a function at specific x-coordinates.

Step 2 - Piecewise Cubic Polynomials:

Divide the data range into smaller subintervals. For each interval [_{xi, xi+1}], construct a cubic polynomial S_i(x) that passes through the endpoints and satisfies certain conditions. These conditions are that the function and its first and second derivatives are continuous at the breakpoints.

Step 3 - Cubic Spline Equations:

For each subinterval [_{xi, xi+1}], you have the cubic spline equation:

S_i(x) = a_i(x - x_i)³ + b_i(x - x_i)² + c_i(x - x_i) + d_i

Step 4 - Coefficient Calculation:

Calculate the coefficients a_i, b_i, c_i, and d_i for each subinterval using the conditions imposed in Step 2. This typically involves solving a system of linear equations.

Step 5 - Evaluate the Cubic Spline:

With the coefficients determined, you can now evaluate the cubic spline S(x) to approximate the function's values at any point x within the data range.

Cubic splines are widely used for smooth interpolation of data points. They provide a piecewise continuous and differentiable approximation, making them a popular choice in computer graphics, data analysis, and other fields.