# File:Tautochrone curve.gif

Tautochrone_curve.gif(300 × 200 pixels, file size: 102 KB, MIME type: image/gif, looped, 80 frames, 3.2 s)

## Summary

Description

A tautochrone curve is the curve for which the time taken by an object sliding without friction in uniform gravity to its lowest point is independent of its starting point. Here, four points at different positions reach the bottom at the same time.

In the graphic, s represents arc length, t represents time, and the blue arrows represent acceleration along the trajectory. As the points reach the horizontal, the velocity becomes constant, the arc length being linear to time.

Date ; new version
Source Own work
Author Claudio Rocchini
rewritten by Geek3
GIF development
 This image was created with Matplotlib.

## Python

```#!/usr/bin/python
# -*- coding: utf8 -*-

'''
animation of balls on a tautochrone curve
'''

import os
import numpy as np
import matplotlib.pyplot as plt
import matplotlib.patches as patches
from matplotlib import animation
from math import *

# settings
fname = 'Tautochrone curve'
width, height = 300, 200
nframes = 80
fps=25

balls = [
{'a':1.0, 'color':'#0000c0'},
{'a':0.8, 'color':'#c00000'},
{'a':0.6, 'color':'#00c000'},
{'a':0.4, 'color':'#c0c000'}]

def curve(phi):
x = phi + sin(phi)
y = 1.0 - cos(phi)
return np.array([x, y])

def animate(nframe, empty=False):
t = nframe / float(nframes - 1.)

# prepare a clean and image-filling canvas for each frame
fig = plt.gcf()
fig.clf()
ax_canvas = plt.gca()
ax_canvas.set_position((0, 0, 1, 1))
ax_canvas.set_xlim(0, width)
ax_canvas.set_ylim(0, height)
ax_canvas.axis('off')

# draw the ramp
x0, y0 = 293, 8
h = 182
npoints = 200
points = []
for i in range(npoints):
phi = i / (npoints - 1.0) * pi - pi
x, y = h/2. * curve(phi) + np.array([x0, y0])
points.append([x, y])

rampline = patches.Polygon(points, closed=False, facecolor='none',
edgecolor='black', linewidth=1.5, capstyle='butt')
points += [[x0-h*pi/2, y0], [x0-h*pi/2, y0+h]]

ramp = patches.Polygon(points, closed=True, facecolor='#c0c0c0', edgecolor='none')

# plot axes
plotw = 0.5
ax_plot = fig.add_axes((0.47, 0.46, plotw, plotw*2/pi*width/height))
ax_plot.set_xlim(0, 1)
ax_plot.set_ylim(0, 1)
for b in balls:
time_array = np.linspace(0, 1, 201)
phi_pendulum_array = (1 - b['a'] * np.cos(time_array*pi/2))
ax_plot.plot(time_array, phi_pendulum_array, '-', color=b['color'], lw=.8)
ax_plot.set_xticks([])
ax_plot.set_yticks([])
ax_plot.set_xlabel('t')
ax_plot.set_ylabel('s')

for b in balls:
# draw the balls
phi_pendulum = b['a'] * -cos(t * pi/2)
phi_wheel = 2 * asin(phi_pendulum)
phi_wheel = -abs(phi_wheel)
x, y = h/2. * curve(phi_wheel) + np.array([x0, y0])
facecolor=b['color'], edgecolor='black'))
ax_plot.plot([t], [1 + phi_pendulum], '.', ms=6., mec='none', mfc='black')

v = h/2. * np.array([1 + cos(phi_wheel), sin(phi_wheel)])
vnorm = v / hypot(v[0], v[1])
# in the harmonic motion, acceleration is proportional to -position
acc_along_line = 38. * -phi_pendulum * vnorm
ax_canvas.arrow(x, y, acc_along_line[0], acc_along_line[1],

fig = plt.figure(figsize=(width/100., height/100.))
print 'saving', fname + '.gif'
#anim = animation.FuncAnimation(fig, animate, frames=nframes)
#anim.save(fname + '.gif', writer='imagemagick', fps=fps)

frames = []
for nframe in range(nframes):
frame = fname + '_{:02}.png'.format(nframe)
animation.FuncAnimation(fig, lambda n: animate(nframe), frames=1).save(
frame, writer='imagemagick')
frames.append(frame)

# assemble animation using imagemagick, this avoids dithering and huge filesize
os.system('convert -delay {} +dither +remap -layers Optimize {} "{}"'.format(
100//fps, ' '.join(['"' + f + '"' for f in frames]), fname + '.gif'))
for frame in frames:
if os.path.exists(frame):
os.remove(frame)
```

## Licensing

I, the copyright holder of this work, hereby publish it under the following licenses:
 Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License.
 This file is licensed under the Creative Commons Attribution 2.5 Generic license. You are free: to share – to copy, distribute and transmit the work to remix – to adapt the work Under the following conditions: attribution – You must attribute the work in the manner specified by the author or licensor (but not in any way that suggests that they endorse you or your use of the work).

## File history

Click on a date/time to view the file as it appeared at that time.

Date/TimeThumbnailDimensionsUserComment
current13:15, 1 August 2009300 × 200 (102 KB)Geek3 (talk | contribs)new physically correct version
06:56, 9 May 2007300 × 200 (602 KB)Rocchini (talk | contribs){{Information |Description=Tautochrone curve animation (4 points runs over a cycloid) |Source=Own work |Date=2007-05-09 |Author=Claudio Rocchini |Permission=CC-BY 2.5 }}
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