Question

Cubic splines. Suppose you are in charge of the design of a roller coaster ride. This simple ride will not make any left or right turns; that is, the track lies in a vertical plane. The accompanying figure shows the ride as viewed from the side. The points $(a_i,b_j)$are given to you, and your job is to connect the dots in a reasonably smooth way. Let ${\mathbf{a}}_{\mathbf{i}\mathbf{+}\mathbf{1}}\mathbf{>}{\mathbf{a}}_{\mathbf{i}\mathbf{}}\mathbf{for}\mathbf{}\mathbf{i}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}\mathbf{n}\mathbf{-}\mathbf{1}$.

One method often employed in such design problems is the technique of cubic splines. We choose ${\mathbf{f}}_{\mathbf{i}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}$, a polynomial of degree $\mathbf{\le }\mathbf{3}$, to define the shape of the ride between $(a_{i-1},b_{i-1})$and $\mathbf{(}{\mathbf{a}}_{\mathbf{i}}\mathbf{,}{\mathbf{b}}_{\mathbf{j}}\mathbf{)}\mathbf{,}\mathbf{}\mathbf{for}\mathbf{}\mathbf{i}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}\mathbf{n}$.

Obviously, it is required that ${\mathbf{f}}_{\mathbf{i}}\left(a_i\right)\mathbf{=}{\mathbf{b}}_{\mathbf{i}}$and ${\mathbf{f}}_{\mathbf{i}}\mathbf{\left(}{\mathbf{a}}_{\mathbf{i}\mathbf{-}\mathbf{1}}\mathbf{\right)}\mathbf{=}{\mathbf{b}}_{\mathbf{i}\mathbf{-}\mathbf{1}}\mathbf{,}\mathbf{}\mathbf{for}\mathbf{}\mathbf{i}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}\mathbf{n}$. To guarantee a smooth ride at the points $\left(a_i,b_i\right)$, we want the first and second derivatives of ${\mathbf{f}}_{\mathbf{i}}$and ${\mathbf{f}}_{\mathbf{i}\mathbf{+}\mathbf{1}}$to agree at these points:

$\mathbf{f}{\mathbf{\text{'}}}_{\mathbf{i}}\mathbf{\left(}{\mathbf{a}}_{\mathbf{i}}\mathbf{\right)}\mathbf{=}\mathbf{f}{\mathbf{\text{'}}}_{\mathbf{i}\mathbf{+}\mathbf{1}}\mathbf{\left(}{\mathbf{a}}_{\mathbf{i}}\mathbf{\right)}$and

$\mathbf{f}\mathbf{\text{'}}{\mathbf{\text{'}}}_{\mathbf{i}}\left(a_i\right)\mathbf{=}\mathbf{f}\mathbf{\text{'}}{\mathbf{\text{'}}}_{\mathbf{i}\mathbf{+}\mathbf{1}}\left(a_i\right)\mathbf{,}\mathbf{}\mathbf{for}\mathbf{}\mathbf{i}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}\mathbf{n}\mathbf{-}\mathbf{1}\mathbf{.}$

Explain the practical significance of these conditions. Explain why, for the convenience of the riders, it is also required that

$\mathbf{f}{\mathbf{\text{'}}}_{\mathbf{1}}\left(a_0\right)\mathbf{=}\mathbf{f}{\mathbf{\text{'}}}_{\mathbf{n}}\left(a_n\right)\mathbf{=}\mathbf{0}$

Show that satisfying all these conditions amounts to solving 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.)

Short answer

For a smoother and non-stop roller coaster ride during specific periods, the condition, $\mathrm{f}{\text{'}}_1\left({\mathrm{a}}_0\right)=\mathrm{f}{\text{'}}_{\mathrm{n}}\left({\mathrm{a}}_{\mathrm{n}}\right)=0$should be satisfied. There are equations and variables in the system.

Step-by-step solution

Consider the condition

To specify the curvature of the roller coaster ride between the points $\left(a_{i-1},b_{i-1}\right)$and $\left(a_i,b_i\right)$, for $i=0,............,n$, we have

$f_i\left(a_i\right)=b_i$, and $f_i\left(a_{i-1}\right)=b_{i-1}........\left(1\right)$

and for the ride to be comfortable at certain points $\left(a_i,b_i\right)$,

 $$\begin{gathered} f{\text{'}}_i\left(a_i\right)=f{\text{'}}_{i+1}\left(a_i\right)......\left(2\right) \\ f\text{'}{\text{'}}_i\left(a_i\right)=f\text{'}{\text{'}}_{i+1}\left(a_i\right)fori=0,.....,n....\left(3\right) \end{gathered}$$

These requirements indicate that the roller coaster ride should be smooth and consistent during the specified periods.

Consider the specification

Also, for the riders' convenience, the ride's beginning and ending points must be parallel to the earth, which means the polynomials at the beginning and ending points must be parallel to the x-axis. Hence,

$f{\text{'}}_1\left(a_0\right)=0andf{\text{'}}_n\left(a_n\right)=0$ and

Final answer

For a smoother and non-stop roller coaster ride during specific periods, the condition $f{\text{'}}_1\left(a_0\right)=f{\text{'}}_n\left(a_n\right)=0$should be satisfied. There are 4n equations and variables in the system.