Two methods of INTERPOLATION

Interpolating polynomial (grey) and cubic spline (blue): Drag the red points with your finger/mouse!
Note the different behavior when you move a point: For the polynomial we get large oscillations far away. For the cubic spline the changes decay away from the point.
The cubic spline mimics a flexible piece of wood: it minimizes the "energy" \(\displaystyle \int_a^b p''(x)^2 dx\).