Spline Curves

Figure 1: Fitting with different degrees of the polynomial.

• There are times when fitting an interpolating polynomial to data points is very difficult. Figure 1a is plot of $f(x)=cos^{10}(x)$ on the interval $[-2, 2]$. It is a nice, smooth curve but has a pronounced maximum at $x=0$ and is near to the $x$-axis for $\vert x\vert > 1$. The curves of Figure 1b,c, d, and e are for polynomials of degrees $-2, -4, -6,$ and $-8$ that match the function at evenly spaced points. None of the polynomials is a good representation of the function.

  Figure 2: Fitting with quadratic in subinterval.

  
• One might think that a solution to the problem would be to break up the interval $[-2, 2]$ into subintervals and fit separate polynomials to the function in these smaller intervals. Figure 2 shows a much better fit if we use a quadratic between $x=-0.65$ and $x=0.65$, with $P(x)=0$ outside that interval. That is better but there are discontinuities in the slope where the separate polynomials join.
• An answer to the dilemma is to use spline curves. Spline curves may be of varying degrees. Suppose that we have a set of $n + 1$ points (which do not have to be evenly spaced):
  
  $$(x_i,y_i),~with ~i=0,1,2,\ldots,n.$$
  
  

• A spline fits a set of $n^{th}$-degree polynomials, $g_i(x)$, between each pair of points, from $x_i$ to $x_{i+1}$. The points at which the splines join are called knots.

  Figure 3: Linear spline.

  
• If the polynomials are all of degree-1, we have a linear spline and the curve would appear as in the Fig. 3. The slopes are discontinuous where the segments join.