Chapter 1.2, Problem 34E

Cubic splines. Suppose you are in charge of the designof a roller coaster ride. This simple ride will not makeany left or right turns; that is, the track lies in a vertical plane. The accompanying figure shows the ride asviewed from the side. The points $\left(a_i,b_i\right)$ are given toyou, and your job is to connect the dots in a reasonablysmooth way. Let $a_{i+1}>a_i,$ for $i=0,\mathrm{...},n-1$ .

One method often employed in such design problems isthe technique of cubic splines. We choose $f_i\left(t\right)$ , a polynomial of degree $\le 3$ , to define the shape of the ridebetween $\left(a_{i-1},b_{i-1}\right)$ and $\left(a_i,b_i\right)$ ,for $i=1,\mathrm{...},n.$ .

Obviously, it is required that $f_i\left(a_i\right)=b_i$ and $f_i\left(a_{i-1}\right)=b_{i-1},$ for $i=1,\mathrm{...},n$ . To guarantee a smooth ride at the points $\left(a_i,b_i\right)$ , we want the first and second derivatives of $f_i$ and $f_{i+1}$ , to agree at these points:

${f^{\prime }}_i\left(a_i\right)={f^{\prime }}_{i+1}\left(a_i\right)$ and ${f^{″}}_i\left(a_i\right)={f^{″}}_{i+1}\left(a_i\right)$ , for $i=1,\mathrm{...},n-1$ .

Explain the practical significance of these conditions. Explain why, for the convenience of the riders, it is also required that ${f^{\prime }}_i\left(a_0\right)={f^{\prime }}_n\left(a_n\right)=0$ .

Show that satisfying all these conditions amounts tosolving a system of linear equations. How many variables are in this system? How many equations? (Note:It can be shown that this system has a unique solution.)