Interpolation Exercise 1



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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

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:



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with np.load('trajectory.npz') as work:
    t=work['t']
    x=work['x']
    y=work['y']



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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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newt=np.linspace(min(t),max(t),200)
aa=interp1d(t,x,kind='cubic')
newx=aa(newt)
ab=interp1d(t,y,kind='cubic')
newy=ab(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:

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.



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plt.plot(x,y,marker='o',linestyle='',label='origninal points')
plt.plot(newx,newy,marker='.',label='Parameterization')
plt.legend();



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assert True # leave this to grade the trajectory plot