Degrees of Freedomwith Natural Cubic Splines
A natural cubic curve of n segments has 4n DOFs
- joining n-1 pairs of points removes n-1 DOFs (leaving 3n+1)
- matching gradients to give C1 continuity removes n-1 DOFs (leaving 2n+2)
- fixing segment end points removes n+1 DOFs (leaving n+1)
- either:
- making curve C2 continuous removes n-1 DOFs (leaving 2)
- defining all gradient removes n+1 DOFs (leaving 0)
Natural cubic splines use up these DOFs either by
- free boundaries: setting end 2nd derivatives to zero
- clamped boundaries: setting the end tangents to any value
But, other types of cubic splines can spend their DOFs differently