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File:Tautochrone curve.gif

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

SummaryEdit

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
Created with Matplotlib-logo.svg
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')
    
    ax_canvas.add_patch(ramp)
    ax_canvas.add_patch(rampline)
    
    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])
        ax_canvas.add_patch(patches.Circle((x, y), radius=6., zorder=3,
                            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],
                 head_width=6, head_length=6, fc='#1b00ff', ec='#1b00ff')

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)

LicensingEdit

I, the copyright holder of this work, hereby publish it under the following licenses:
GNU head 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.
w:en:Creative Commons
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  • to share – to copy, distribute and transmit the work
  • to remix – to adapt the work
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  • 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).
  • share alike – If you alter, transform, or build upon this work, you may distribute the resulting work only under the same or similar license to this one.
This licensing tag was added to this file as part of the GFDL licensing update.
w:en:Creative Commons
attribution
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).

You may select the license of your choice.

File history

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

Date/TimeThumbnailDimensionsUserComment
current13:15, 1 August 2009Thumbnail for version as of 13:15, 1 August 2009300 × 200 (102 KB)Geek3 (talk | contribs)new physically correct version
06:56, 9 May 2007Thumbnail for version as of 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 }}
  • You cannot overwrite this file.

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