Interpolation Exercise 1



In [1]:

    
%matplotlib inline
import matplotlib.pyplot as plt
import seaborn as sns
import numpy as np



In [2]:

    
from scipy.interpolate import interp1d

2D trajectory interpolation

The file trajectory.npz contains 3 Numpy arrays that describe a 2d trajectory of a particle as a function of time:

t which has discrete values of time t[i].
x which has values of the x position at those times: x[i] = x(t[i]).
x which has values of the y position at those times: y[i] = y(t[i]).

Load those arrays into this notebook and save them as variables x, y and t:



In [3]:

    
dictionary = np.load('trajectory.npz')



In [4]:

    
y = dictionary.items()[0][1]
t = dictionary.items()[1][1]
x = dictionary.items()[2][1]



In [5]:

    
assert isinstance(x, np.ndarray) and len(x)==40
assert isinstance(y, np.ndarray) and len(y)==40
assert isinstance(t, np.ndarray) and len(t)==40

Use these arrays to create interpolated functions $x(t)$ and $y(t)$. Then use those functions to create the following arrays:

newt which has 200 points between $\{t_{min},t_{max}\}$.
newx which has the interpolated values of $x(t)$ at those times.
newy which has the interpolated values of $y(t)$ at those times.



In [6]:

    
x_approx = interp1d(t, x, kind='cubic')
y_approx = interp1d(t, y, kind='cubic')

newt = np.linspace(0,4,200)
newx = x_approx(newt)
newy = y_approx(newt)



In [7]:

    
assert newt[0]==t.min()
assert newt[-1]==t.max()
assert len(newt)==200
assert len(newx)==200
assert len(newy)==200

Make a parametric plot of $\{x(t),y(t)\}$ that shows the interpolated values and the original points:

For the interpolated points, use a solid line.
For the original points, use circles of a different color and no line.
Customize you plot to make it effective and beautiful.



In [11]:

    
plt.figure(figsize=(12,8));
plt.plot(x, y, marker='o', linestyle='', label='Original Data')
plt.plot(newx, newy, label='Interpolated Curve');
plt.legend();
plt.xlabel('X(t)');
plt.ylabel('Y(t)');
plt.title('Position as a Function of Time');



In [9]:

    
assert True # leave this to grade the trajectory plot