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]:
f = np.load("trajectory.npz")
t = f['t']
x = f['x']
y = f['y']

In [4]:
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 [5]:
X = interp1d(t, x, kind='cubic')
Y = interp1d(t, y, kind='cubic')
newt = np.linspace(min(t), max(t), 200)
newx = X(newt)
newy = Y(newt)

In [6]:
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 [8]:
plt.plot(t, x, 'b', linestyle='', marker='o', label="Original Point for x(t)")
plt.plot(t, y, 'r', linestyle = '', marker='o', label="Original Point for y(t)")
plt.plot(newt, newx, 'c-', label="Interpoled x(t)")
plt.plot(newt, newy, 'm-', label="Interpoled y(t)")
ax = plt.gca()
plt.xlabel("Time", fontsize=14)
plt.title("X and Y Trajectory", fontsize = 14)
plt.ylabel("Position", fontsize = 14)

In [59]:
assert True # leave this to grade the trajectory plot

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