Solving a system of linear equations

Question

Assignment Problem: 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_i) are given to you, and your job is to connect the dots in area reasonably smooth way.

Let a_{i+ 1} > a_i, for i = 0,.......,n-1.

One method often employed in such design problems is the technique of cubic splines. We choose f_i(t), a polynomial of degree ≤ 3, to define the shape of the ride between (a_i-i, b_i-i) and (a_i, b_i), for i = 1, . . . , n

Obviously, it is required that f_i(a_i) = b_i and f_i(a_i-1)= b_i-1. for i = 1, . . . , n. To guarantee a smooth ride at the points (a_i, b_i), we want the first and second derivatives of f_i' and f_i+1 to agree at these points:

f_'i(a_i) = f'_i+1(a_i) and

f"_i (a_i) = fi"_i+1(a_i), for i = 1, . . . , n - 1.

Explain the practical significance of these conditions.

Explain why, for the convenience of the riders, it is also required that

f'₁(a₀) = f'_n(a_n) = 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?

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