Animated plots

Animations are generated from a list (or other iterable) of graphics objects. Images are produced by calling the save_image method on each input object, creating a sequence of PNG files. These are then assembled to various target formats using different tools. In particular, the magick/convert program from ImageMagick can be used to generate an animated GIF file. FFmpeg (with the command line program ffmpeg) provides support for various video formats, but also an alternative method of generating animated GIFs. For browsers which support it, APNG can be used as another alternative which works without any extra dependencies.

Warning

Note that ImageMagick and FFmpeg are not included with Sage, and must be installed by the user. On unix systems, type which magick at a command prompt to see if magick (part of the ImageMagick suite) is installed. If it is, you will be given its location. Similarly, you can check for ffmpeg with which ffmpeg. See the websites of ImageMagick or FFmpeg for installation instructions.

EXAMPLES:

The sine function:

sage: x = SR.var("x")
sage: sines = [plot(c*sin(x), (-2*pi,2*pi), color=Color(c,0,0), ymin=-1, ymax=1)
....:          for c in sxrange(0,1,.2)]
sage: a = animate(sines)
sage: print(a)
Animation with 5 frames
sage: a.show()                  # long time  # optional -- ImageMagick

>>> from sage.all import *
>>> x = SR.var("x")
>>> sines = [plot(c*sin(x), (-Integer(2)*pi,Integer(2)*pi), color=Color(c,Integer(0),Integer(0)), ymin=-Integer(1), ymax=Integer(1))
...          for c in sxrange(Integer(0),Integer(1),RealNumber('.2'))]
>>> a = animate(sines)
>>> print(a)
Animation with 5 frames
>>> a.show()                  # long time  # optional -- ImageMagick

x = SR.var("x")
sines = [plot(c*sin(x), (-2*pi,2*pi), color=Color(c,0,0), ymin=-1, ymax=1)
         for c in sxrange(0,1,.2)]
a = animate(sines)
print(a)
a.show()                  # long time  # optional -- ImageMagick

Animate using FFmpeg instead of ImageMagick:

sage: a.show(use_ffmpeg=True)   # long time  # optional -- FFmpeg

>>> from sage.all import *
>>> a.show(use_ffmpeg=True)   # long time  # optional -- FFmpeg

a.show(use_ffmpeg=True)   # long time  # optional -- FFmpeg

Animate as an APNG:

sage: a.apng(show_path=True)    # long time
Animation saved to ....png.

>>> from sage.all import *
>>> a.apng(show_path=True)    # long time
Animation saved to ....png.

a.apng(show_path=True)    # long time

An animated sage.plot.multigraphics.GraphicsArray of rotating ellipses:

sage: E = animate((graphics_array([[ellipse((0,0), a, b, angle=t, xmin=-3, xmax=3)
....:                                + circle((0,0), 3, color='blue')
....:                               for a in range(1,3)]
....:                              for b in range(2,4)])
....:              for t in sxrange(0, pi/4, .15)))
sage: str(E)    # animations produced from a generator do not have a known length
'Animation with unknown number of frames'
sage: E.show()                  # long time  # optional -- ImageMagick

>>> from sage.all import *
>>> E = animate((graphics_array([[ellipse((Integer(0),Integer(0)), a, b, angle=t, xmin=-Integer(3), xmax=Integer(3))
...                                + circle((Integer(0),Integer(0)), Integer(3), color='blue')
...                               for a in range(Integer(1),Integer(3))]
...                              for b in range(Integer(2),Integer(4))])
...              for t in sxrange(Integer(0), pi/Integer(4), RealNumber('.15'))))
>>> str(E)    # animations produced from a generator do not have a known length
'Animation with unknown number of frames'
>>> E.show()                  # long time  # optional -- ImageMagick

E = animate((graphics_array([[ellipse((0,0), a, b, angle=t, xmin=-3, xmax=3)
                               + circle((0,0), 3, color='blue')
                              for a in range(1,3)]
                             for b in range(2,4)])
             for t in sxrange(0, pi/4, .15)))
str(E)    # animations produced from a generator do not have a known length
E.show()                  # long time  # optional -- ImageMagick

A simple animation of a circle shooting up to the right:

sage: c = animate([circle((i,i), 1 - 1/(i+1), hue=i/10)
....:              for i in srange(0, 2, 0.2)],
....:             xmin=0, ymin=0, xmax=2, ymax=2, figsize=[2,2])
sage: c.show()                  # long time  # optional -- ImageMagick

>>> from sage.all import *
>>> c = animate([circle((i,i), Integer(1) - Integer(1)/(i+Integer(1)), hue=i/Integer(10))
...              for i in srange(Integer(0), Integer(2), RealNumber('0.2'))],
...             xmin=Integer(0), ymin=Integer(0), xmax=Integer(2), ymax=Integer(2), figsize=[Integer(2),Integer(2)])
>>> c.show()                  # long time  # optional -- ImageMagick

c = animate([circle((i,i), 1 - 1/(i+1), hue=i/10)
             for i in srange(0, 2, 0.2)],
            xmin=0, ymin=0, xmax=2, ymax=2, figsize=[2,2])
c.show()                  # long time  # optional -- ImageMagick

Animations of 3d objects:

sage: s,t = SR.var("s,t")
sage: def sphere_and_plane(x):
....:     return (sphere((0,0,0), 1, color='red', opacity=.5)
....:              + parametric_plot3d([t,x,s], (s,-1,1), (t,-1,1),
....:                                  color='green', opacity=.7))
sage: sp = animate([sphere_and_plane(x)
....:               for x in sxrange(-1, 1, .3)])
sage: sp[0]      # first frame
Graphics3d Object
sage: sp[-1]     # last frame
Graphics3d Object
sage: sp.show()                 # long time  # optional -- ImageMagick

sage: (x,y,z) = SR.var("x,y,z")
sage: def frame(t):
....:     return implicit_plot3d((x^2 + y^2 + z^2),
....:                            (x, -2, 2), (y, -2, 2), (z, -2, 2),
....:                            plot_points=60, contour=[1,3,5],
....:                            region=lambda x,y,z: x<=t or y>=t or z<=t)
sage: a = animate([frame(t) for t in srange(.01, 1.5, .2)])
sage: a[0]                      # long time
Graphics3d Object
sage: a.show()                  # long time  # optional -- ImageMagick

>>> from sage.all import *
>>> s,t = SR.var("s,t")
>>> def sphere_and_plane(x):
...     return (sphere((Integer(0),Integer(0),Integer(0)), Integer(1), color='red', opacity=RealNumber('.5'))
...              + parametric_plot3d([t,x,s], (s,-Integer(1),Integer(1)), (t,-Integer(1),Integer(1)),
...                                  color='green', opacity=RealNumber('.7')))
>>> sp = animate([sphere_and_plane(x)
...               for x in sxrange(-Integer(1), Integer(1), RealNumber('.3'))])
>>> sp[Integer(0)]      # first frame
Graphics3d Object
>>> sp[-Integer(1)]     # last frame
Graphics3d Object
>>> sp.show()                 # long time  # optional -- ImageMagick

>>> (x,y,z) = SR.var("x,y,z")
>>> def frame(t):
...     return implicit_plot3d((x**Integer(2) + y**Integer(2) + z**Integer(2)),
...                            (x, -Integer(2), Integer(2)), (y, -Integer(2), Integer(2)), (z, -Integer(2), Integer(2)),
...                            plot_points=Integer(60), contour=[Integer(1),Integer(3),Integer(5)],
...                            region=lambda x,y,z: x<=t or y>=t or z<=t)
>>> a = animate([frame(t) for t in srange(RealNumber('.01'), RealNumber('1.5'), RealNumber('.2'))])
>>> a[Integer(0)]                      # long time
Graphics3d Object
>>> a.show()                  # long time  # optional -- ImageMagick

s,t = SR.var("s,t")
def sphere_and_plane(x):
    return (sphere((0,0,0), 1, color='red', opacity=.5)
             + parametric_plot3d([t,x,s], (s,-1,1), (t,-1,1),
                                 color='green', opacity=.7))
sp = animate([sphere_and_plane(x)
              for x in sxrange(-1, 1, .3)])
sp[0]      # first frame
sp[-1]     # last frame
sp.show()                 # long time  # optional -- ImageMagick
(x,y,z) = SR.var("x,y,z")
def frame(t):
    return implicit_plot3d((x^2 + y^2 + z^2),
                           (x, -2, 2), (y, -2, 2), (z, -2, 2),
                           plot_points=60, contour=[1,3,5],
                           region=lambda x,y,z: x<=t or y>=t or z<=t)
a = animate([frame(t) for t in srange(.01, 1.5, .2)])
a[0]                      # long time
a.show()                  # long time  # optional -- ImageMagick

If the input objects do not have a save_image method, then the animation object attempts to make an image by calling its internal method sage.plot.animate.Animation.make_image(). This is illustrated by the following example:

sage: t = SR.var("t")
sage: a = animate((sin(c*pi*t) for c in sxrange(1, 2, .2)))
sage: a.show()                  # long time  # optional -- ImageMagick

>>> from sage.all import *
>>> t = SR.var("t")
>>> a = animate((sin(c*pi*t) for c in sxrange(Integer(1), Integer(2), RealNumber('.2'))))
>>> a.show()                  # long time  # optional -- ImageMagick

t = SR.var("t")
a = animate((sin(c*pi*t) for c in sxrange(1, 2, .2)))
a.show()                  # long time  # optional -- ImageMagick

AUTHORS:

  • William Stein

  • John Palmieri

  • Niles Johnson (2013-12): Expand to animate more graphics objects

  • Martin von Gagern (2014-12): Added APNG support

  • Joshua Campbell (2020): interactive animation via Three.js viewer

REFERENCES:

class sage.plot.animate.APngAssembler(out, num_frames, num_plays=0, delay=200, delay_denominator=100)[source]

Bases: object

Build an APNG (Animated PNG) from a sequence of PNG files. This is used by the sage.plot.animate.Animation.apng() method.

This code is quite simple; it does little more than copying chunks from input PNG files to the output file. There is no optimization involved. This does not depend on external programs or libraries.

INPUT:

  • out – a file opened for binary writing to which the data will be written

  • num_frames – the number of frames in the animation

  • num_plays – how often to iterate, 0 means infinitely

  • delay – numerator of the delay fraction in seconds

  • delay_denominator – denominator of the delay in seconds

EXAMPLES:

sage: from sage.plot.animate import APngAssembler
sage: x = SR.var("x")
sage: def assembleAPNG():
....:     a = animate([sin(x + float(k)) for k in srange(0,2*pi,0.7)],
....:                 xmin=0, xmax=2*pi, figsize=[2,1])
....:     pngdir = a.png()
....:     outfile = sage.misc.temporary_file.tmp_filename(ext='.png')
....:     with open(outfile, "wb") as f:
....:         apng = APngAssembler(f, len(a))
....:         for i in range(len(a)):
....:             png = os.path.join(pngdir, "{:08d}.png".format(i))
....:             apng.add_frame(png, delay=10*i + 10)
....:     return outfile
sage: assembleAPNG()  # long time
'...png'

>>> from sage.all import *
>>> from sage.plot.animate import APngAssembler
>>> x = SR.var("x")
>>> def assembleAPNG():
...     a = animate([sin(x + float(k)) for k in srange(Integer(0),Integer(2)*pi,RealNumber('0.7'))],
...                 xmin=Integer(0), xmax=Integer(2)*pi, figsize=[Integer(2),Integer(1)])
...     pngdir = a.png()
...     outfile = sage.misc.temporary_file.tmp_filename(ext='.png')
...     with open(outfile, "wb") as f:
...         apng = APngAssembler(f, len(a))
...         for i in range(len(a)):
...             png = os.path.join(pngdir, "{:08d}.png".format(i))
...             apng.add_frame(png, delay=Integer(10)*i + Integer(10))
...     return outfile
>>> assembleAPNG()  # long time
'...png'

from sage.plot.animate import APngAssembler
x = SR.var("x")
def assembleAPNG():
    a = animate([sin(x + float(k)) for k in srange(0,2*pi,0.7)],
                xmin=0, xmax=2*pi, figsize=[2,1])
    pngdir = a.png()
    outfile = sage.misc.temporary_file.tmp_filename(ext='.png')
    with open(outfile, "wb") as f:
        apng = APngAssembler(f, len(a))
        for i in range(len(a)):
            png = os.path.join(pngdir, "{:08d}.png".format(i))
            apng.add_frame(png, delay=10*i + 10)
    return outfile
assembleAPNG()  # long time
add_frame(pngfile, delay=None, delay_denominator=None)[source]

Add a single frame to the APNG file.

INPUT:

  • pngfile – file name of the PNG file with data for this frame

  • delay – numerator of the delay fraction in seconds

  • delay_denominator – denominator of the delay in seconds

If the delay is not specified, the default from the constructor applies.

magic = b'\x89PNG\r\n\x1a\n'
mustmatch = frozenset({b'IHDR', b'PLTE', b'bKGD', b'cHRM', b'gAMA', b'pHYs', b'sBIT', b'tRNS'})
set_default(pngfile)[source]

Add a default image for the APNG file.

This image is used as a fallback in case some application does not understand the APNG format. This method must be called prior to any calls to the add_frame method, if it is called at all. If it is not called, then the first frame of the animation will be the default.

INPUT:

  • pngfile – file name of the PNG file with data for the default image

class sage.plot.animate.Animation(v=None, **kwds)[source]

Bases: WithEqualityById, SageObject

Return an animation of a sequence of plots of objects.

INPUT:

  • v – iterable of Sage objects; these should preferably be graphics objects, but if they aren’t, then make_image() is called on them.

  • xmin, xmax, ymin, ymax – the ranges of the x and y axes

  • **kwds – all additional inputs are passed onto the rendering command. E.g., use figsize to adjust the resolution and aspect ratio.

EXAMPLES:

sage: x = SR.var("x")
sage: a = animate([sin(x + float(k)) for k in srange(0, 2*pi, 0.3)],
....:             xmin=0, xmax=2*pi, figsize=[2,1])
sage: print(a)
Animation with 21 frames
sage: print(a[:5])
Animation with 5 frames
sage: a.show()              # long time  # optional -- ImageMagick
sage: a[:5].show()          # long time  # optional -- ImageMagick

>>> from sage.all import *
>>> x = SR.var("x")
>>> a = animate([sin(x + float(k)) for k in srange(Integer(0), Integer(2)*pi, RealNumber('0.3'))],
...             xmin=Integer(0), xmax=Integer(2)*pi, figsize=[Integer(2),Integer(1)])
>>> print(a)
Animation with 21 frames
>>> print(a[:Integer(5)])
Animation with 5 frames
>>> a.show()              # long time  # optional -- ImageMagick
>>> a[:Integer(5)].show()          # long time  # optional -- ImageMagick

x = SR.var("x")
a = animate([sin(x + float(k)) for k in srange(0, 2*pi, 0.3)],
            xmin=0, xmax=2*pi, figsize=[2,1])
print(a)
print(a[:5])
a.show()              # long time  # optional -- ImageMagick
a[:5].show()          # long time  # optional -- ImageMagick

The show() method takes arguments to specify the delay between frames (measured in hundredths of a second, default value 20) and the number of iterations (default: 0, which means to iterate forever). To iterate 4 times with half a second between each frame:

sage: a.show(delay=50, iterations=4)  # long time  # optional -- ImageMagick

>>> from sage.all import *
>>> a.show(delay=Integer(50), iterations=Integer(4))  # long time  # optional -- ImageMagick

a.show(delay=50, iterations=4)  # long time  # optional -- ImageMagick

An animation of drawing a parabola:

sage: step = 0.1
sage: L = Graphics()
sage: v = []
sage: for i in srange(0, 1, step):
....:     L += line([(i,i^2),(i+step,(i+step)^2)], rgbcolor=(1,0,0), thickness=2)
....:     v.append(L)
sage: a = animate(v, xmin=0, ymin=0)
sage: a.show()              # long time  # optional -- ImageMagick
sage: show(L)

>>> from sage.all import *
>>> step = RealNumber('0.1')
>>> L = Graphics()
>>> v = []
>>> for i in srange(Integer(0), Integer(1), step):
...     L += line([(i,i**Integer(2)),(i+step,(i+step)**Integer(2))], rgbcolor=(Integer(1),Integer(0),Integer(0)), thickness=Integer(2))
...     v.append(L)
>>> a = animate(v, xmin=Integer(0), ymin=Integer(0))
>>> a.show()              # long time  # optional -- ImageMagick
>>> show(L)

step = 0.1
L = Graphics()
v = []
for i in srange(0, 1, step):
    L += line([(i,i^2),(i+step,(i+step)^2)], rgbcolor=(1,0,0), thickness=2)
    v.append(L)
a = animate(v, xmin=0, ymin=0)
a.show()              # long time  # optional -- ImageMagick
show(L)
apng(savefile=None, show_path=False, delay=20, iterations=0)[source]

Create an animated PNG composed from rendering the graphics objects in self. Return the absolute path to that file.

Notice that not all web browsers are capable of displaying APNG files, though they should still present the first frame of the animation as a fallback.

The generated file is not optimized, so it may be quite large.

Input:

  • delay – (default: 20) delay in hundredths of a second between frames

  • savefile – file that the animated gif gets saved to

  • iterations – integer (default: 0); number of iterations of animation. If 0, loop forever.

  • show_path – boolean (default: False); if True, print the path to the saved file

EXAMPLES:

sage: x = SR.var("x")
sage: a = animate([sin(x + float(k))
....:              for k in srange(0,2*pi,0.7)],
....:             xmin=0, xmax=2*pi, figsize=[2,1])
sage: dir = tmp_dir()
sage: a.apng(show_path=True)  # long time
Animation saved to ....png.
sage: a.apng(savefile=dir + 'my_animation.png', delay=35, iterations=3)  # long time
sage: a.apng(savefile=dir + 'my_animation.png', show_path=True)  # long time
Animation saved to .../my_animation.png.

>>> from sage.all import *
>>> x = SR.var("x")
>>> a = animate([sin(x + float(k))
...              for k in srange(Integer(0),Integer(2)*pi,RealNumber('0.7'))],
...             xmin=Integer(0), xmax=Integer(2)*pi, figsize=[Integer(2),Integer(1)])
>>> dir = tmp_dir()
>>> a.apng(show_path=True)  # long time
Animation saved to ....png.
>>> a.apng(savefile=dir + 'my_animation.png', delay=Integer(35), iterations=Integer(3))  # long time
>>> a.apng(savefile=dir + 'my_animation.png', show_path=True)  # long time
Animation saved to .../my_animation.png.

x = SR.var("x")
a = animate([sin(x + float(k))
             for k in srange(0,2*pi,0.7)],
            xmin=0, xmax=2*pi, figsize=[2,1])
dir = tmp_dir()
a.apng(show_path=True)  # long time
a.apng(savefile=dir + 'my_animation.png', delay=35, iterations=3)  # long time
a.apng(savefile=dir + 'my_animation.png', show_path=True)  # long time

If the individual frames have different sizes, an error will be raised:

sage: a = animate([plot(sin(x), (x, 0, k))
....:              for k in range(1,4)],
....:             ymin=-1, ymax=1, aspect_ratio=1, figsize=[2,1])
sage: a.apng()  # long time
Traceback (most recent call last):
...
ValueError: Chunk IHDR mismatch

>>> from sage.all import *
>>> a = animate([plot(sin(x), (x, Integer(0), k))
...              for k in range(Integer(1),Integer(4))],
...             ymin=-Integer(1), ymax=Integer(1), aspect_ratio=Integer(1), figsize=[Integer(2),Integer(1)])
>>> a.apng()  # long time
Traceback (most recent call last):
...
ValueError: Chunk IHDR mismatch

a = animate([plot(sin(x), (x, 0, k))
             for k in range(1,4)],
            ymin=-1, ymax=1, aspect_ratio=1, figsize=[2,1])
a.apng()  # long time
ffmpeg(savefile=None, show_path=False, output_format=None, ffmpeg_options='', delay=None, iterations=0, pix_fmt='rgb24')[source]

Return a movie showing an animation composed from rendering the frames in self.

This method will only work if ffmpeg is installed. See https://www.ffmpeg.org for information about ffmpeg.

INPUT:

  • savefile – file that the mpeg gets saved to

Warning

This will overwrite savefile if it already exists.

  • show_path – boolean (default: False); if True, print the path to the saved file

  • output_format – string (default: None); format and suffix to use for the video. This may be 'mpg', 'mpeg', 'avi', 'gif', or any other format that ffmpeg can handle. If this is None and the user specifies savefile with a suffix, say savefile='animation.avi', try to determine the format ('avi' in this case) from that file name. If no file is specified or if the suffix cannot be determined, 'mpg' is used.

  • ffmpeg_options – string (default: ''); this string is passed directly to ffmpeg

  • delay – integer (default: None); delay in hundredths of a second between frames. The framerate is 100/delay. This is not supported for mpeg files: for mpegs, the frame rate is always 25 fps.

  • iterations – integer (default: 0); number of iterations of animation. If 0, loop forever. This is only supported for animated gif output and requires ffmpeg version 0.9 or later. For older versions, set iterations=None.

  • pix_fmt – string (default: 'rgb24'); used only for gif output. Different values such as ‘rgb8’ or ‘pal8’ may be necessary depending on how ffmpeg was installed. Set pix_fmt=None to disable this option.

If savefile is not specified: in notebook mode, display the animation; otherwise, save it to a default file name. Use sage.misc.verbose.set_verbose() with level=1 to see additional output.

EXAMPLES:

sage: x = SR.var("x")
sage: a = animate([sin(x + float(k))
....:              for k in srange(0, 2*pi, 0.7)],
....:             xmin=0, xmax=2*pi, ymin=-1, ymax=1, figsize=[2,1])
sage: td = tmp_dir()
sage: a.ffmpeg(savefile=td + 'new.mpg')                  # long time  # optional -- FFmpeg
sage: a.ffmpeg(savefile=td + 'new.avi')                  # long time  # optional -- FFmpeg
sage: a.ffmpeg(savefile=td + 'new.gif')                  # long time  # optional -- FFmpeg
sage: a.ffmpeg(savefile=td + 'new.mpg', show_path=True)  # long time  # optional -- FFmpeg
Animation saved to .../new.mpg.

>>> from sage.all import *
>>> x = SR.var("x")
>>> a = animate([sin(x + float(k))
...              for k in srange(Integer(0), Integer(2)*pi, RealNumber('0.7'))],
...             xmin=Integer(0), xmax=Integer(2)*pi, ymin=-Integer(1), ymax=Integer(1), figsize=[Integer(2),Integer(1)])
>>> td = tmp_dir()
>>> a.ffmpeg(savefile=td + 'new.mpg')                  # long time  # optional -- FFmpeg
>>> a.ffmpeg(savefile=td + 'new.avi')                  # long time  # optional -- FFmpeg
>>> a.ffmpeg(savefile=td + 'new.gif')                  # long time  # optional -- FFmpeg
>>> a.ffmpeg(savefile=td + 'new.mpg', show_path=True)  # long time  # optional -- FFmpeg
Animation saved to .../new.mpg.

x = SR.var("x")
a = animate([sin(x + float(k))
             for k in srange(0, 2*pi, 0.7)],
            xmin=0, xmax=2*pi, ymin=-1, ymax=1, figsize=[2,1])
td = tmp_dir()
a.ffmpeg(savefile=td + 'new.mpg')                  # long time  # optional -- FFmpeg
a.ffmpeg(savefile=td + 'new.avi')                  # long time  # optional -- FFmpeg
a.ffmpeg(savefile=td + 'new.gif')                  # long time  # optional -- FFmpeg
a.ffmpeg(savefile=td + 'new.mpg', show_path=True)  # long time  # optional -- FFmpeg

Note

If ffmpeg is not installed, you will get an error message like this:

FeatureNotPresentError: ffmpeg is not available.
Executable 'ffmpeg' not found on PATH.
Further installation instructions might be available at https://www.ffmpeg.org/.
gif(delay=20, savefile=None, iterations=0, show_path=False, use_ffmpeg=False)[source]

Return an animated gif composed from rendering the graphics objects in self.

This method will only work if either (a) the ImageMagick software suite is installed, i.e., you have the magick/convert command or (b) ffmpeg is installed. See the web sites of ImageMagick and FFmpeg for more details. By default, this produces the gif using Imagemagick if it is present. If this can’t find ImageMagick or if use_ffmpeg is True, then it uses ffmpeg instead.

INPUT:

  • delay – (default: 20) delay in hundredths of a second between frames

  • savefile – file that the animated gif gets saved to

  • iterations – integer (default: 0); number of iterations of animation. If 0, loop forever.

  • show_path – boolean (default: False); if True, print the path to the saved file

  • use_ffmpeg – boolean (default: False); if True, use ‘ffmpeg’ by default instead of ImageMagick

If savefile is not specified: in notebook mode, display the animation; otherwise, save it to a default file name.

EXAMPLES:

sage: x = SR.var("x")
sage: a = animate([sin(x + float(k))
....:              for k in srange(0,2*pi,0.7)],
....:             xmin=0, xmax=2*pi, ymin=-1, ymax=1, figsize=[2,1])
sage: td = tmp_dir()
sage: a.gif()              # not tested
sage: a.gif(savefile=td + 'my_animation.gif',               # long time  # optional -- ImageMagick
....:       delay=35, iterations=3)
sage: with open(td + 'my_animation.gif', 'rb') as f:        # long time  # optional -- ImageMagick
....:     print(b'GIF8' in f.read())
True
sage: a.gif(savefile=td + 'my_animation.gif',               # long time  # optional -- ImageMagick
....:       show_path=True)
Animation saved to .../my_animation.gif.
sage: a.gif(savefile=td + 'my_animation_2.gif',             # long time  # optional -- FFmpeg
....:       show_path=True, use_ffmpeg=True)
Animation saved to .../my_animation_2.gif.

>>> from sage.all import *
>>> x = SR.var("x")
>>> a = animate([sin(x + float(k))
...              for k in srange(Integer(0),Integer(2)*pi,RealNumber('0.7'))],
...             xmin=Integer(0), xmax=Integer(2)*pi, ymin=-Integer(1), ymax=Integer(1), figsize=[Integer(2),Integer(1)])
>>> td = tmp_dir()
>>> a.gif()              # not tested
>>> a.gif(savefile=td + 'my_animation.gif',               # long time  # optional -- ImageMagick
...       delay=Integer(35), iterations=Integer(3))
>>> with open(td + 'my_animation.gif', 'rb') as f:        # long time  # optional -- ImageMagick
...     print(b'GIF8' in f.read())
True
>>> a.gif(savefile=td + 'my_animation.gif',               # long time  # optional -- ImageMagick
...       show_path=True)
Animation saved to .../my_animation.gif.
>>> a.gif(savefile=td + 'my_animation_2.gif',             # long time  # optional -- FFmpeg
...       show_path=True, use_ffmpeg=True)
Animation saved to .../my_animation_2.gif.

x = SR.var("x")
a = animate([sin(x + float(k))
             for k in srange(0,2*pi,0.7)],
            xmin=0, xmax=2*pi, ymin=-1, ymax=1, figsize=[2,1])
td = tmp_dir()
a.gif()              # not tested
a.gif(savefile=td + 'my_animation.gif',               # long time  # optional -- ImageMagick
      delay=35, iterations=3)
with open(td + 'my_animation.gif', 'rb') as f:        # long time  # optional -- ImageMagick
    print(b'GIF8' in f.read())
a.gif(savefile=td + 'my_animation.gif',               # long time  # optional -- ImageMagick
      show_path=True)
a.gif(savefile=td + 'my_animation_2.gif',             # long time  # optional -- FFmpeg
      show_path=True, use_ffmpeg=True)

Note

If neither ffmpeg nor ImageMagick is installed, you will get an error message like this:

Error: Neither ImageMagick nor ffmpeg appears to be installed. Saving an
animation to a GIF file or displaying an animation requires one of these
packages, so please install one of them and try again.

See www.imagemagick.org and www.ffmpeg.org for more information.
graphics_array(ncols=3)[source]

Return a sage.plot.multigraphics.GraphicsArray with plots of the frames of this animation, using the given number of columns. The frames must be acceptable inputs for sage.plot.multigraphics.GraphicsArray.

EXAMPLES:

sage: # needs sage.schemes
sage: E = EllipticCurve('37a')
sage: v = [E.change_ring(GF(p)).plot(pointsize=30)
....:      for p in [97, 101, 103]]
sage: a = animate(v, xmin=0, ymin=0, axes=False)
sage: print(a)
Animation with 3 frames
sage: a.show()                      # optional -- ImageMagick

>>> from sage.all import *
>>> # needs sage.schemes
>>> E = EllipticCurve('37a')
>>> v = [E.change_ring(GF(p)).plot(pointsize=Integer(30))
...      for p in [Integer(97), Integer(101), Integer(103)]]
>>> a = animate(v, xmin=Integer(0), ymin=Integer(0), axes=False)
>>> print(a)
Animation with 3 frames
>>> a.show()                      # optional -- ImageMagick

# needs sage.schemes
E = EllipticCurve('37a')
v = [E.change_ring(GF(p)).plot(pointsize=30)
     for p in [97, 101, 103]]
a = animate(v, xmin=0, ymin=0, axes=False)
print(a)
a.show()                      # optional -- ImageMagick

Modify the default arrangement of array:

sage: g = a.graphics_array(); print(g)                                      # needs sage.schemes
Graphics Array of size 1 x 3
sage: g.show(figsize=[6,3])                                                 # needs sage.schemes

>>> from sage.all import *
>>> g = a.graphics_array(); print(g)                                      # needs sage.schemes
Graphics Array of size 1 x 3
>>> g.show(figsize=[Integer(6),Integer(3)])                                                 # needs sage.schemes

g = a.graphics_array(); print(g)                                      # needs sage.schemes
g.show(figsize=[6,3])                                                 # needs sage.schemes

Specify different arrangement of array and save it with a given file name:

sage: g = a.graphics_array(ncols=2); print(g)                               # needs sage.schemes
Graphics Array of size 2 x 2
sage: f = tmp_filename(ext='.png'); print(f)                                # needs sage.schemes
...png
sage: g.save(f)                                                             # needs sage.schemes

>>> from sage.all import *
>>> g = a.graphics_array(ncols=Integer(2)); print(g)                               # needs sage.schemes
Graphics Array of size 2 x 2
>>> f = tmp_filename(ext='.png'); print(f)                                # needs sage.schemes
...png
>>> g.save(f)                                                             # needs sage.schemes

g = a.graphics_array(ncols=2); print(g)                               # needs sage.schemes
f = tmp_filename(ext='.png'); print(f)                                # needs sage.schemes
g.save(f)                                                             # needs sage.schemes

Frames can be specified as a generator too; it is internally converted to a list:

sage: t = SR.var("t")
sage: b = animate((plot(sin(c*pi*t)) for c in sxrange(1,2,.2)))
sage: g = b.graphics_array()
sage: print(g)
Graphics Array of size 2 x 3

>>> from sage.all import *
>>> t = SR.var("t")
>>> b = animate((plot(sin(c*pi*t)) for c in sxrange(Integer(1),Integer(2),RealNumber('.2'))))
>>> g = b.graphics_array()
>>> print(g)
Graphics Array of size 2 x 3

t = SR.var("t")
b = animate((plot(sin(c*pi*t)) for c in sxrange(1,2,.2)))
g = b.graphics_array()
print(g)
interactive(**kwds)[source]

Create an interactive depiction of the animation.

INPUT:

  • **kwds – any of the viewing options accepted by show() are valid as keyword arguments to this function and they will behave in the same way. Those that are animation-related and recognized by the Three.js viewer are: animate, animation_controls, auto_play, delay, and loop.

OUTPUT: a 3D graphics object which, by default, will use the Three.js viewer

EXAMPLES:

sage: x = SR.var("x")
sage: frames = [point3d((sin(x), cos(x), x))
....:           for x in (0, pi/16, .., 2*pi)]
sage: animate(frames).interactive(online=True)
Graphics3d Object

>>> from sage.all import *
>>> x = SR.var("x")
>>> frames = [point3d((sin(x), cos(x), x))
...           for x in (ellipsis_iter(Integer(0), pi/Integer(16),Ellipsis, Integer(2)*pi))]
>>> animate(frames).interactive(online=True)
Graphics3d Object

x = SR.var("x")
frames = [point3d((sin(x), cos(x), x))
          for x in (0, pi/16, .., 2*pi)]
animate(frames).interactive(online=True)

Works with frames that are 2D or 3D graphics objects or convertible to 2D or 3D graphics objects via a plot or plot3d method:

sage: frames = [dodecahedron(), circle(center=(0, 0), radius=1), x^2]
sage: animate(frames).interactive(online=True, delay=100)
Graphics3d Object

>>> from sage.all import *
>>> frames = [dodecahedron(), circle(center=(Integer(0), Integer(0)), radius=Integer(1)), x**Integer(2)]
>>> animate(frames).interactive(online=True, delay=Integer(100))
Graphics3d Object

frames = [dodecahedron(), circle(center=(0, 0), radius=1), x^2]
animate(frames).interactive(online=True, delay=100)

See also

Three.js JavaScript WebGL Renderer

make_image(frame, filename, **kwds)[source]

Given a frame which has no save_image() method, make a graphics object and save it as an image with the given filename. By default, this is sage.plot.plot.plot(). To make animations of other objects, override this method in a subclass.

EXAMPLES:

sage: from sage.plot.animate import Animation
sage: class MyAnimation(Animation):
....:    def make_image(self, frame, filename, **kwds):
....:        P = parametric_plot(frame[0], frame[1], **frame[2])
....:        P.save_image(filename, **kwds)

sage: t = SR.var("t")
sage: x = lambda t: cos(t)
sage: y = lambda n,t: sin(t)/n
sage: B = MyAnimation([([x(t), y(i+1,t)], (t,0,1),
....:                   {'color':Color((1,0,i/4)), 'aspect_ratio':1, 'ymax':1})
....:                  for i in range(4)])

sage: d = B.png(); v = os.listdir(d); v.sort(); v  # long time
['00000000.png', '00000001.png', '00000002.png', '00000003.png']
sage: B.show()  # not tested

sage: class MyAnimation(Animation):
....:    def make_image(self, frame, filename, **kwds):
....:        G = frame.plot()
....:        G.set_axes_range(floor(G.xmin()), ceil(G.xmax()),
....:                         floor(G.ymin()), ceil(G.ymax()))
....:        G.save_image(filename, **kwds)

sage: B = MyAnimation([graphs.CompleteGraph(n)
....:                  for n in range(7,11)], figsize=5)
sage: d = B.png()
sage: v = os.listdir(d); v.sort(); v
['00000000.png', '00000001.png', '00000002.png', '00000003.png']
sage: B.show()  # not tested

>>> from sage.all import *
>>> from sage.plot.animate import Animation
>>> class MyAnimation(Animation):
...    def make_image(self, frame, filename, **kwds):
...        P = parametric_plot(frame[Integer(0)], frame[Integer(1)], **frame[Integer(2)])
...        P.save_image(filename, **kwds)

>>> t = SR.var("t")
>>> x = lambda t: cos(t)
>>> y = lambda n,t: sin(t)/n
>>> B = MyAnimation([([x(t), y(i+Integer(1),t)], (t,Integer(0),Integer(1)),
...                   {'color':Color((Integer(1),Integer(0),i/Integer(4))), 'aspect_ratio':Integer(1), 'ymax':Integer(1)})
...                  for i in range(Integer(4))])

>>> d = B.png(); v = os.listdir(d); v.sort(); v  # long time
['00000000.png', '00000001.png', '00000002.png', '00000003.png']
>>> B.show()  # not tested

>>> class MyAnimation(Animation):
...    def make_image(self, frame, filename, **kwds):
...        G = frame.plot()
...        G.set_axes_range(floor(G.xmin()), ceil(G.xmax()),
...                         floor(G.ymin()), ceil(G.ymax()))
...        G.save_image(filename, **kwds)

>>> B = MyAnimation([graphs.CompleteGraph(n)
...                  for n in range(Integer(7),Integer(11))], figsize=Integer(5))
>>> d = B.png()
>>> v = os.listdir(d); v.sort(); v
['00000000.png', '00000001.png', '00000002.png', '00000003.png']
>>> B.show()  # not tested

from sage.plot.animate import Animation
class MyAnimation(Animation):
   def make_image(self, frame, filename, **kwds):
       P = parametric_plot(frame[0], frame[1], **frame[2])
       P.save_image(filename, **kwds)
t = SR.var("t")
x = lambda t: cos(t)
y = lambda n,t: sin(t)/n
B = MyAnimation([([x(t), y(i+1,t)], (t,0,1),
                  {'color':Color((1,0,i/4)), 'aspect_ratio':1, 'ymax':1})
                 for i in range(4)])
d = B.png(); v = os.listdir(d); v.sort(); v  # long time
B.show()  # not tested
class MyAnimation(Animation):
   def make_image(self, frame, filename, **kwds):
       G = frame.plot()
       G.set_axes_range(floor(G.xmin()), ceil(G.xmax()),
                        floor(G.ymin()), ceil(G.ymax()))
       G.save_image(filename, **kwds)
B = MyAnimation([graphs.CompleteGraph(n)
                 for n in range(7,11)], figsize=5)
d = B.png()
v = os.listdir(d); v.sort(); v
B.show()  # not tested
png(dir=None)[source]

Render PNG images of the frames in this animation, saving them in dir. Return the absolute path to that directory. If the frames have been previously rendered and dir is None, just return the directory in which they are stored.

When dir is other than None, force re-rendering of frames.

INPUT:

  • dir – (default: None) directory in which to store frames; in this case, a temporary directory will be created for storing the frames

OUTPUT: absolute path to the directory containing the PNG images

EXAMPLES:

sage: x = SR.var("x")
sage: a = animate([plot(x^2 + n) for n in range(4)], ymin=0, ymax=4)
sage: d = a.png(); v = os.listdir(d); v.sort(); v  # long time
['00000000.png', '00000001.png', '00000002.png', '00000003.png']

>>> from sage.all import *
>>> x = SR.var("x")
>>> a = animate([plot(x**Integer(2) + n) for n in range(Integer(4))], ymin=Integer(0), ymax=Integer(4))
>>> d = a.png(); v = os.listdir(d); v.sort(); v  # long time
['00000000.png', '00000001.png', '00000002.png', '00000003.png']

x = SR.var("x")
a = animate([plot(x^2 + n) for n in range(4)], ymin=0, ymax=4)
d = a.png(); v = os.listdir(d); v.sort(); v  # long time
save(filename=None, show_path=False, use_ffmpeg=False, **kwds)[source]

Save this animation.

INPUT:

  • filename – (default: None) name of save file

  • show_path – boolean (default: False); if True, print the path to the saved file

  • use_ffmpeg – boolean (default: False); if True, use ‘ffmpeg’ by default instead of ImageMagick when creating GIF files

If filename is None, then in notebook mode, display the animation; otherwise, save the animation to a GIF file. If filename ends in ‘.html’, save an interactive() version of the animation to an HTML file that uses the Three.js viewer. If filename ends in ‘.sobj’, save to an sobj file. Otherwise, try to determine the format from the filename extension (‘.mpg’, ‘.gif’, ‘.avi’, etc.). If the format cannot be determined, default to GIF.

For GIF files, either ffmpeg or the ImageMagick suite must be installed. For other movie formats, ffmpeg must be installed. sobj and HTML files can be saved with no extra software installed.

EXAMPLES:

sage: x = SR.var("x")
sage: a = animate([sin(x + float(k))
....:              for k in srange(0, 2*pi, 0.7)],
....:             xmin=0, xmax=2*pi, ymin=-1, ymax=1, figsize=[2,1])
sage: td = tmp_dir()
sage: a.save()         # not tested
sage: a.save(td + 'wave.gif')                   # long time  # optional -- ImageMagick
sage: a.save(td + 'wave.gif', show_path=True)   # long time  # optional -- ImageMagick
Animation saved to file .../wave.gif.
sage: a.save(td + 'wave.avi', show_path=True)   # long time  # optional -- FFmpeg
Animation saved to file .../wave.avi.
sage: a.save(td + 'wave0.sobj')
sage: a.save(td + 'wave1.sobj', show_path=True)
Animation saved to file .../wave1.sobj.
sage: a.save(td + 'wave0.html', online=True)
sage: a.save(td + 'wave1.html', show_path=True, online=True)
Animation saved to file .../wave1.html.

>>> from sage.all import *
>>> x = SR.var("x")
>>> a = animate([sin(x + float(k))
...              for k in srange(Integer(0), Integer(2)*pi, RealNumber('0.7'))],
...             xmin=Integer(0), xmax=Integer(2)*pi, ymin=-Integer(1), ymax=Integer(1), figsize=[Integer(2),Integer(1)])
>>> td = tmp_dir()
>>> a.save()         # not tested
>>> a.save(td + 'wave.gif')                   # long time  # optional -- ImageMagick
>>> a.save(td + 'wave.gif', show_path=True)   # long time  # optional -- ImageMagick
Animation saved to file .../wave.gif.
>>> a.save(td + 'wave.avi', show_path=True)   # long time  # optional -- FFmpeg
Animation saved to file .../wave.avi.
>>> a.save(td + 'wave0.sobj')
>>> a.save(td + 'wave1.sobj', show_path=True)
Animation saved to file .../wave1.sobj.
>>> a.save(td + 'wave0.html', online=True)
>>> a.save(td + 'wave1.html', show_path=True, online=True)
Animation saved to file .../wave1.html.

x = SR.var("x")
a = animate([sin(x + float(k))
             for k in srange(0, 2*pi, 0.7)],
            xmin=0, xmax=2*pi, ymin=-1, ymax=1, figsize=[2,1])
td = tmp_dir()
a.save()         # not tested
a.save(td + 'wave.gif')                   # long time  # optional -- ImageMagick
a.save(td + 'wave.gif', show_path=True)   # long time  # optional -- ImageMagick
a.save(td + 'wave.avi', show_path=True)   # long time  # optional -- FFmpeg
a.save(td + 'wave0.sobj')
a.save(td + 'wave1.sobj', show_path=True)
a.save(td + 'wave0.html', online=True)
a.save(td + 'wave1.html', show_path=True, online=True)
show(delay=None, iterations=None, **kwds)[source]

Show this animation immediately.

This method attempts to display the graphics immediately, without waiting for the currently running code (if any) to return to the command line. Be careful, calling it from within a loop will potentially launch a large number of external viewer programs.

INPUT:

  • delay – (default: 20) delay in hundredths of a second between frames

  • iterations – integer (default: 0); number of iterations of animation. If 0, loop forever.

  • format – (default: gif) format to use for output. Currently supported formats are: gif, ogg, webm, mp4, flash, matroska, avi, wmv, quicktime.

OUTPUT:

This method does not return anything. Use save() if you want to save the figure as an image.

Note

Currently this is done using an animated gif, though this could change in the future. This requires that either ffmpeg or the ImageMagick suite (in particular, the magick/convert command) is installed.

See also the ffmpeg() method.

EXAMPLES:

sage: x = SR.var("x")
sage: a = animate([sin(x + float(k))
....:              for k in srange(0,2*pi,0.7)],
....:             xmin=0, xmax=2*pi, figsize=[2,1])
sage: a.show()                              # long time  # optional -- ImageMagick

>>> from sage.all import *
>>> x = SR.var("x")
>>> a = animate([sin(x + float(k))
...              for k in srange(Integer(0),Integer(2)*pi,RealNumber('0.7'))],
...             xmin=Integer(0), xmax=Integer(2)*pi, figsize=[Integer(2),Integer(1)])
>>> a.show()                              # long time  # optional -- ImageMagick

x = SR.var("x")
a = animate([sin(x + float(k))
             for k in srange(0,2*pi,0.7)],
            xmin=0, xmax=2*pi, figsize=[2,1])
a.show()                              # long time  # optional -- ImageMagick

The preceding will loop the animation forever. If you want to show only three iterations instead:

sage: a.show(iterations=3)                  # long time  # optional -- ImageMagick

>>> from sage.all import *
>>> a.show(iterations=Integer(3))                  # long time  # optional -- ImageMagick

a.show(iterations=3)                  # long time  # optional -- ImageMagick

To put a half-second delay between frames:

sage: a.show(delay=50)                      # long time  # optional -- ImageMagick

>>> from sage.all import *
>>> a.show(delay=Integer(50))                      # long time  # optional -- ImageMagick

a.show(delay=50)                      # long time  # optional -- ImageMagick

You can also make use of the HTML5 video element in the Sage Notebook:

sage: # long time, optional -- FFmpeg
sage: a.show(format='ogg')
sage: a.show(format='webm')
sage: a.show(format='mp4')
sage: a.show(format='webm', iterations=1)

>>> from sage.all import *
>>> # long time, optional -- FFmpeg
>>> a.show(format='ogg')
>>> a.show(format='webm')
>>> a.show(format='mp4')
>>> a.show(format='webm', iterations=Integer(1))

# long time, optional -- FFmpeg
a.show(format='ogg')
a.show(format='webm')
a.show(format='mp4')
a.show(format='webm', iterations=1)

Other backends may support other file formats as well:

sage: # long time, optional -- FFmpeg
sage: a.show(format='flash')
sage: a.show(format='matroska')
sage: a.show(format='avi')
sage: a.show(format='wmv')
sage: a.show(format='quicktime')

>>> from sage.all import *
>>> # long time, optional -- FFmpeg
>>> a.show(format='flash')
>>> a.show(format='matroska')
>>> a.show(format='avi')
>>> a.show(format='wmv')
>>> a.show(format='quicktime')

# long time, optional -- FFmpeg
a.show(format='flash')
a.show(format='matroska')
a.show(format='avi')
a.show(format='wmv')
a.show(format='quicktime')

Note

If you don’t have ffmpeg or ImageMagick installed, you will get an error message like this:

Error: Neither ImageMagick nor ffmpeg appears to be installed. Saving an
animation to a GIF file or displaying an animation requires one of these
packages, so please install one of them and try again.

See www.imagemagick.org and www.ffmpeg.org for more information.
sage.plot.animate.animate(frames, **kwds)[source]

Animate a list of frames by creating a sage.plot.animate.Animation object.

EXAMPLES:

sage: t = SR.var("t")
sage: a = animate((cos(c*pi*t) for c in sxrange(1, 2, .2)))
sage: a.show()              # long time  # optional -- ImageMagick

>>> from sage.all import *
>>> t = SR.var("t")
>>> a = animate((cos(c*pi*t) for c in sxrange(Integer(1), Integer(2), RealNumber('.2'))))
>>> a.show()              # long time  # optional -- ImageMagick

t = SR.var("t")
a = animate((cos(c*pi*t) for c in sxrange(1, 2, .2)))
a.show()              # long time  # optional -- ImageMagick

See also sage.plot.animate for more examples.