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Plotting random points inside parabola using Python?

Was wondering how to generate 100 random points inside the parabola below and also ensure that these points have y values smaller than 15000?

x = np.linspace(-100,100)
y = 2*x**2+10*x

plt.plot(x,y)
plt.show()

plot

Any help would be appreciated. Thanks!

Solution

  • enter image description here

    The usual boilerplate

    import numpy as np
import matplotlib.pyplot as plt

    We want to generate points in a rectangle (x0, x0+dx) × (y0, y0+dy), and discard points under the parabola.

    The y range is easy, y0=0 and dy=15000 y0=ymin=y(-2.5), dy=15000-ymin, for the x range we need to solve 2 x² + 10 x = 15000 ⇒ x₁ = (-5 - √(5²+2·15000))/2 and x₂ = (-5 + √(5²+2·15000))/2.

    x0, dx = (-5-np.sqrt(30025))/2, np.sqrt(30025)
y0, dy = 2*6.25+10*(-2.5), 15000.0-(2*6.25+10*(-2.5))

    Now we generate a list of random points, that is initially void

    xy = []

    Then, we generate the coordinates of a point, check if it's valid and repeat until we have accumulated 100 valid points.

    while len(xy) < 100:
    x = x0 + np.random.rand()*dx
    y = np.random.rand()*15000
    if y<2*x*x+10*x : continue # i.e., start a new iteration w/o appending
    xy.append((x,y))

    Now, the plotting, I'll use a range that is symmetrical wrt the axis of symmetry of the parabola, then I plot the points using scatter (note the *zip(*iterable) that is used to transpose a 2D list) and finally I draw a horizontal dashed line.

    x = np.linspace(-2.5-100, -2.5+100, 801)
y = 2*x*(x+5)

plt.plot(x, y)
plt.scatter(*zip(*xy), s=1)
plt.hlines(15000, x[0]-5, x[-1]+5, linestyles='dashed')

plt.show()