Generating Random Smooth Curves Using Physics Principles

Here is a general process for making random smooth curves using physics principles:

1) Initialize a particle at the origin with some random velocity.
2) Apply a constant force which is orthogonal to the velocity.
3) Numerically integrate to calculate and plot the trajectory
4) Change the magnitude / sign of the force periodically, while maintaining the orthogonality

This ensures that the particle maintains a constant speed while continuously and smoothly changing its direction.

Plots and source code are below:

import math
import random

x = 0.0
y = 0.0

m = 1.0

vx = -10.0 + 20.0 * random.random()
vy = -10.0 + 20.0 * random.random()

ax = 0.0
ay = 0.0

t = 0.0
dt = 10.0**(-2.0)

count = 0

Fbase = 50.0

Fmag = -Fbase + 2.0 * Fbase * random.random()

while t <= 100.0:

    x = x + vx * dt
    y = y + vy * dt

    vx = vx + ax * dt
    vy = vy + ay * dt

    if (count % 100) == 0:

        Fmag = -Fbase + 2.0 * Fbase * random.random()

    Nx = -vy
    Ny = vx

    Nmag = math.hypot(vx,vy)

    Nx = Nx / Nmag
    Ny = Ny / Nmag

    Fx = Fmag * Nx
    Fy = Fmag * Ny

    ax = Fx / m
    ay = Fy / m

    print t,x,y

    t = t + dt
    count = count + 1

Note by Steven Chase
4 months, 1 week ago

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what visualization software is that ?

Andrew Hucek - 3 months, 4 weeks ago

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I'm very primitive. I use Python to print out (x,y) coordinate pairs. And then I paste into Excel and make a scatter plot.

Steven Chase - 3 months, 4 weeks ago

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have you tried what happens with increasing the range ?

Andrew Hucek - 3 months, 4 weeks ago

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@Andrew Hucek You mean the run time?

Steven Chase - 3 months, 4 weeks ago

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@Steven Chase I probably did, and thought that this was some sort of optimum. I actually wrote the code for this a year ago or so. It might be fun to try a spherical coordinates version too.

Steven Chase - 3 months, 4 weeks ago

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@Steven Chase yes exactly, also thought about it. Btw, do you use python 2 or 3 ?

Andrew Hucek - 3 months, 4 weeks ago

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@Andrew Hucek I use Python 2.7

Steven Chase - 3 months, 4 weeks ago

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@Steven Chase yes, bad vocab, sorry

Andrew Hucek - 3 months, 4 weeks ago

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quite impressive just for that :o

Andrew Hucek - 3 months, 4 weeks ago

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This must have taken you forever

Annie Li - 4 months, 1 week ago

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About an hour

Steven Chase - 4 months, 1 week ago

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Wow. You must like coding a lot

Annie Li - 4 months, 1 week ago

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