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Suppose we are fitting three points P 0, P 1, P 2 with two cubic splines. Prove that if 3( p 2 − p 0) = v 0 + 4 v 1 + v 2, then the second derivatives are equal at the point where the splines meet. Your formulas should involve p 0, p 1, p 2, v 0, v 1, v 2. Find p′ 12( t) for the second spline at P 1.įind formulas for the two second-derivative vectors found in part (a). In Example 5.24, find the second-derivative vector p′ 01( t) of the first spline at P 1. What system of equations involving the polynomial coefficients do you obtain? (b)Ĭan you solve this system and find the formula for p (t)? CURVE FITTING WITH CUBIC SPLINESįit the following data with two cubic splines: P 0 = (2, 5), P 1 = (3, 7), P 2 = (4, 13), V 0 =(1, 1), V 1 = (1, 4), V 2 = (1, 6). Have given tangent vectors v 0 and v 1 at these points, and (iii) Pass through given points P 0 and P 1, (ii) Suppose you want a fourth-degree vector polynomial p ( t) to 9.įind the single cubic spline for the following data: P 0 = (1, 1), V 0 = 〈0, l〉, P i = (2, 3), V 1 = (1, 0). Can you do it? 8.įind the single cubic spline for the following data: P 0 = (2, −4), v 0 = 〈1, −l〉, P i = (5, 1), v i = (2, 0). Suppose you wanted to fit the four points in part (a) of Exercise 6 with a single cubic vector polynomial. How would you answer part (a) if the last point were changed to (4, 7)? 7. If you can't, explain what goes wrong when you try. (a) Suppose you wanted to fit the following four points with a single quadratic vector polynomial: P 0(1, 4), P 1(2, 3), P 2(3, 4), P 3(4, 8), passing through P i when t = i. But what happens if we try to fit them with a single quadratic vector polynomial? Work this out. The points P 0(2, 5), P 1(3, 9), P 2(4, 13) lie on a line that is, they could be fit with a single linear vector polynomial p (t) = c 0t + c 1 passing through P i when t = i. 4.įit the following three points with a single quadratic vector polynomial: At t = 0 it should pass through P 0(2, −3), at t = 1 through P 1(3, −1), at t = 2 through P 2(4, −3). Fit the following three points with a single quadratic vector polynomial: At t = 0 it should pass through P 0(2, 5), at t = 1 through P 1(3, 7), at t = 2 through P 2(4, 13).