Knots and B-Splines: Understanding Knots Simplified

[phiby]

I have been reading about B-Splines for a couple of days now, but I just can't get what a knot is! I have referred multiple books & websites. Can someone explain knots to me a in a simple language?

 

[HallsofIvy]

If you have read about splines, then presumably you know that they are piecewise polynomial functions. "Knots" are the points where the "pieces" change. That is, where the formula changes.

If f(x) is the "quadratic spline" that is given by
f(x)= x^2 for 0\le x\le 2, f(x)= 2x^2- 4x+ 4 for 2\le x\le 3, etc. then x= 2 or, more properly, the point (2, 4) is a "knot".

Typically one wants a spline that interpolates given values. Often the knots are chosen at the "interpolation points" but not always.

 

[phiby]

"Knots" are the points where the "pieces" change. That is, where the formula changes.

Assume a B-spline with n+1 control points (0 to n) & d control points per curve(polynomial of degree (d-1)). The text says that B-Splines are defined only in the interval from knot value u_d-1 to u_n+1. So what are the extra knots at the beginning and end of the knot vector for?

 

[HallsofIvy]

Probably just to give additional equations. For example, with cubic splines, you have 4 coefficients per cubic and can require that the values of the function and first and second derivatives match at each knot. If you have n knots, including the two ends, you have two values requiring that both sides give the correct value there and two more equations requiring that the first and second derivatives match, for a total of 4 equations, at the n-2 interior knots but only one equation at the two ends for a total of 4(n-2)+ 2= 4n- 6 equations. But you also have n-1 intervals between those n points so a total of 4n- 4 coefficient to solve for. That is two fewer equations than coefficents!

What you can do is add the "not a knot" condition- you require that the polynomials in the first two intervals at each end be exactly the same. That is, that two "knots" are, in fact, not knots. That gives the value of the function at those two points as the two more equations you need.