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%matplotlib inline
import matplotlib.pyplot as plt
import seaborn as sns
import numpy as np
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from scipy.interpolate import interp1d, interp2d
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:
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f = np.load('trajectory.npz')
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x = f['x']
y = f['y']
t = f['t']
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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.
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x_approx = interp1d(t, x, kind='cubic')
y_approx = interp1d(t, y, kind='cubic')
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traj_approx = interp2d(x, y, t, kind='cubic')
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newt = np.linspace(t.min(),max(t),200)
newx = x_approx(newt)
newy = y_approx(newt)
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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:
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fig = plt.figure(figsize=(7,7))
plt.plot(newx, newy, marker='.')
plt.plot(x, y, 'ro')
plt.xticks([-1.0,-0.5,0.0,0.5,1.0])
plt.yticks([-1.0,-0.5,0.0,0.5,1.0])
plt.xlabel('x(t)')
plt.ylabel('y(t)')
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assert True # leave this to grade the trajectory plot