Why don’t we learn about quadratic spline? Is it not as useful as cubic spline?
This is a good question. I mentioned something in class, but I can say more here.
In short, we do not use quadratic splines for teaching purposes, because
they would require different data points from knots, which may complicate
To say a bit more, recall that, on n subintervals (n+1 knots),
linear C^0 splines have n+1 free parameters,
quadratic C^1 splines have n+2 parameters,
cubic C^2 splines have n+3 parameters,
and so on.
Linear splines use the n+1 knots as data points for interpolation
(and these are enough to uniquely determine them),
cubic splines use the n+1 knots as data points for interpolation
and another two end-conditions to make up n+3 equations.
Quadratic splines would need n+2 data points for interpolation.
It is hard to make these data points balanced, if the n+1 knots are used.
(One extra condition would be needed from one only end.)
Unbalanced conditions lead to instablilities.
So, quadratic splines use the n midpoints of subintervals
and the two end points, to make n+2 equations for interpolation.
This makes the presentation a bit more complicated.
So, quadratic splines are used, but not for teaching.