Ch19.4: Cubic Splines

What's brown and sticky?

A stick.

Wanna hear a joke about paper?

Never mind-it's tearable.

• Instead of trying to fit a piecewise linear model through a set of data points, we can try to fit a piecewise cubic polynomial model through the points.
  
• For a smooth curve (no jaggedness), we require that the resulting graph is differentiable everywhere.

• Suppose that we want to fit cubic splines through the data points (1,2), (2,1), (3,2).
  
• Find two cubic polynomials (splines) that fit their respective two data points along with additional smoothness conditions.

\[ \begin{aligned} p_1(x) &= a_1 x^3 + b_1 x^2 + c_1 x + d_1 , \,\, 1 \leq x \leq 2 \\ p_2(x) &= a_2 x^3 + b_2 x^2 + c_2 x + d_2 , \,\, 2 \leq x \leq 3 \end{aligned} \]

• Here, \( p_1(x) \) passes through (1,2) and (2,1).
• Also, \( p_2(x) \) passes through (2,1) and (3,2).
• To join smoothly at the shared point (2,1), they need to have the same slope and concavity there.

Find a spline approximation and a polynomial approximation for the curve of the cross section of the circular shaped Shrine of the Book in Jerusalem.

import numpy as np
import matplotlib.pyplot as plt
from scipy.interpolate import interp1d

#Enter data
x = [-5.8, -5.0, -4.0, -2.5, -1.5, -0.8, 0, 0.8, 1.5, 2.5, 4.0, 5.0, 5.8]
y = [0, 1.5, 1.8, 2.2, 2.7, 3.5, 3.9, 3.5, 2.7, 2.2, 1.8, 1.5, 0]

#Command for cubic spline spline polynomial p(x)
p = interp1d(x, y, kind = 'cubic')

#Create vector of 500 points between 0 and x[n-1] on x-axis
n = len(x)
xnew = np.linspace(x[0], x[n-1], num=500, endpoint=True)

#Plot the original data together with p(x) sampled at the 500 points
plt.plot(x, y, 'o', xnew, p(xnew), '-,r')
plt.legend(['Original Data','Cubic Spline p(x)'], loc = 'best')
plt.show()