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r""" The Tachyon 3D Ray Tracer
Given any 3D graphics object one can compute a raytraced representation by typing ``show(viewer='tachyon')``. For example, we draw two translucent spheres that contain a red tube, and render the result using Tachyon.
::
sage: S = sphere(opacity=0.8, aspect_ratio=[1,1,1]) sage: L = line3d([(0,0,0),(2,0,0)], thickness=10, color='red') sage: M = S + S.translate((2,0,0)) + L sage: M.show(viewer='tachyon')
One can also directly control Tachyon, which gives a huge amount of flexibility. For example, here we directly use Tachyon to draw 3 spheres on the coordinate axes::
sage: t = Tachyon(xres=500,yres=500, camera_center=(2,0,0)) sage: t.light((4,3,2), 0.2, (1,1,1)) sage: t.texture('t2', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1,0,0)) sage: t.texture('t3', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(0,1,0)) sage: t.texture('t4', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(0,0,1)) sage: t.sphere((0,0.5,0), 0.2, 't2') sage: t.sphere((0.5,0,0), 0.2, 't3') sage: t.sphere((0,0,0.5), 0.2, 't4') sage: t.show()
For scenes with many reflections it is helpful to increase the raydepth option, and turn on antialiasing. The following scene is an extreme case with many reflections between four cotangent spheres::
sage: t = Tachyon(camera_center=(0,-4,1), xres = 800, yres = 600, raydepth = 12, aspectratio=.75, antialiasing = 4) sage: t.light((0.02,0.012,0.001), 0.01, (1,0,0)) sage: t.light((0,0,10), 0.01, (0,0,1)) sage: t.texture('s', color = (.8,1,1), opacity = .9, specular = .95, diffuse = .3, ambient = 0.05) sage: t.texture('p', color = (0,0,1), opacity = 1, specular = .2) sage: t.sphere((-1,-.57735,-0.7071),1,'s') sage: t.sphere((1,-.57735,-0.7071),1,'s') sage: t.sphere((0,1.15465,-0.7071),1,'s') sage: t.sphere((0,0,0.9259),1,'s') sage: t.plane((0,0,-1.9259),(0,0,1),'p') sage: t.show() # long time
Different projection options are available. The following examples all use a sphere and cube::
sage: cedges = [[[1, 1, 1], [-1, 1, 1]], [[1, 1, 1], [1, -1, 1]], ....: [[1, 1, 1], [1, 1, -1]], [[-1, 1, 1], [-1, -1, 1]], [[-1, 1, 1], ....: [-1, 1, -1]], [[1, -1, 1], [-1, -1, 1]], [[1, -1, 1], [1, -1, -1]], ....: [[-1, -1, 1], [-1, -1, -1]], [[1, 1, -1], [-1, 1, -1]], ....: [[1, 1, -1], [1, -1, -1]], [[-1, 1, -1], [-1, -1, -1]], ....: [[1, -1, -1], [-1, -1, -1]]]
The default projection is ``'perspective'``::
sage: t = Tachyon(xres=800, yres=600, camera_center=(-1.5,0.0,0.0), zoom=.2) sage: t.texture('t1', color=(0,0,1)) sage: for ed in cedges: ....: t.fcylinder(ed[0], ed[1], .05, 't1') sage: t.light((-4,-4,4), .1, (1,1,1)) sage: t.show()
Another option is ``projection='fisheye'``, which requires frustrum information. The frustrum data is (bottom angle, top angle, left angle, right angle)::
sage: t = Tachyon(xres=800, yres=600, camera_center=(-1.5,0.0,0.0), ....: projection='fisheye', frustum=(-1.2, 1.2, -1.2, 1.2)) sage: t.texture('t1', color=(0,0,1)) sage: for ed in cedges: ....: t.fcylinder(ed[0], ed[1], .05, 't1') sage: t.light((-4,-4,4), .1, (1,1,1)) sage: t.show()
Finally there is the ``projection='perspective_dof'`` option. ::
sage: T = Tachyon(xres=800, antialiasing=4, raydepth=10, ....: projection='perspective_dof', focallength='1.0', aperture='.0025') sage: T.light((0,5,7), 1.0, (1,1,1)) sage: T.texture('t1', opacity=1, specular=.3) sage: T.texture('t2', opacity=1, specular=.3, color=(0,0,1)) sage: T.texture('t3', opacity=1, specular=1, color=(1,.8,1), diffuse=0.2) sage: T.plane((0,0,-1), (0,0,1), 't3') sage: ttlist = ['t1', 't2'] sage: tt = 't1' sage: T.cylinder((0,0,.1), (1,1/3,0), .05, 't3') sage: for q in srange(-3, 100, .15): ....: if tt == 't1': ....: tt = 't2' ....: else: ....: tt = 't1' ....: T.sphere((q, q/3+.3*sin(3*q), .1+.3*cos(3*q)), .1, tt) sage: T.show()
Image files in the ``ppm`` format can be used to tile planes or cover cylinders or spheres. In this example an image is created and then used to tile the plane::
sage: T = Tachyon(xres=800, yres=600, camera_center=(-2.0,-.1,.3), projection='fisheye', frustum=(-1.0, 1.0, -1.0, 1.0)) sage: T.texture('t1',color=(0,0,1)) sage: for ed in cedges: ....: T.fcylinder(ed[0], ed[1], .05, 't1') sage: T.light((-4,-4,4),.1,(1,1,1)) sage: fname_png = tmp_filename(ext='.png') sage: fname_ppm = tmp_filename(ext='.ppm') sage: T.save(fname_png) sage: r2 = os.system('convert '+fname_png+' '+fname_ppm) # optional -- ImageMagick
sage: T = Tachyon(xres=800, yres=600, camera_center=(-2.0,-.1,.3), projection='fisheye', frustum=(-1.0, 1.0, -1.0, 1.0)) # optional -- ImageMagick sage: T.texture('t1', color=(1,0,0), specular=.9) # optional -- ImageMagick sage: T.texture('p1', color=(1,1,1), opacity=.1, imagefile=fname_ppm, texfunc=9) # optional -- ImageMagick sage: T.sphere((0,0,0), .5, 't1') # optional -- ImageMagick sage: T.plane((0,0,-1), (0,0,1), 'p1') # optional -- ImageMagick sage: T.light((-4,-4,4), .1, (1,1,1)) # optional -- ImageMagick sage: T.show() # optional -- ImageMagick
AUTHOR:
- John E. Stone (johns@megapixel.com): wrote tachyon ray tracer
- William Stein: sage-tachyon interface
- Joshua Kantor: 3d function plotting
- Tom Boothby: 3d function plotting n'stuff
- Leif Hille: key idea for bugfix for texfunc issue (:trac:`799`)
- Marshall Hampton: improved doctests, rings, axis-aligned boxes.
- Paul Graham: Respect global verbosity settings (:trac:`16228`)
.. TODO::
- clean up trianglefactory stuff """ from __future__ import absolute_import
from .tri_plot import Triangle, SmoothTriangle, TriangleFactory, TrianglePlot
from sage.interfaces.tachyon import tachyon_rt
from sage.misc.fast_methods import WithEqualityById from sage.structure.sage_object import SageObject
from sage.misc.misc import SAGE_TMP from sage.misc.misc import get_verbose from sage.misc.viewer import png_viewer from sage.misc.temporary_file import tmp_filename
#from sage.ext import fast_tachyon_routines
import os
from math import sqrt
class Tachyon(WithEqualityById, SageObject): r""" Create a scene the can be rendered using the Tachyon ray tracer.
INPUT:
- ``xres`` - (default 350) - ``yres`` - (default 350) - ``zoom`` - (default 1.0) - ``antialiasing`` - (default ``False``) - ``aspectratio`` - (default 1.0) - ``raydepth`` - (default 5) - ``camera_center`` - (default (-3, 0, 0)) - ``updir`` - (default (0, 0, 1)) - ``look_at`` - (default (0,0,0)) - ``viewdir`` - (default ``None``), otherwise list of three numbers - ``projection`` - ``'PERSPECTIVE'`` (default), ``'perspective_dof'`` or ``'fisheye'``. - ``frustum`` - (default ''), otherwise list of four numbers. Only used with projection='fisheye'. - ``focallength`` - (default ''), otherwise a number. Only used with projection='perspective_dof'. - ``aperture`` - (default ''), otherwise a number. Only used with projection='perspective_dof'.
OUTPUT: A Tachyon 3d scene.
Note that the coordinates are by default such that `z` is up, positive `y` is to the {left} and `x` is toward you. This is not oriented according to the right hand rule.
EXAMPLES: Spheres along the twisted cubic.
::
sage: t = Tachyon(xres=512,yres=512, camera_center=(3,0.3,0)) sage: t.light((4,3,2), 0.2, (1,1,1)) sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0)) sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0)) sage: t.texture('t2', ambient=0.2,diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0)) sage: k=0 sage: for i in srange(-1,1,0.05): ....: k += 1 ....: t.sphere((i,i^2-0.5,i^3), 0.1, 't%s'%(k%3)) sage: t.show()
Another twisted cubic, but with a white background, got by putting infinite planes around the scene.
::
sage: t = Tachyon(xres=512,yres=512, camera_center=(3,0.3,0), raydepth=8) sage: t.light((4,3,2), 0.2, (1,1,1)) sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0)) sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0)) sage: t.texture('t2', ambient=0.2,diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0)) sage: t.texture('white', color=(1,1,1)) sage: t.plane((0,0,-1), (0,0,1), 'white') sage: t.plane((0,-20,0), (0,1,0), 'white') sage: t.plane((-20,0,0), (1,0,0), 'white')
::
sage: k=0 sage: for i in srange(-1,1,0.05): ....: k += 1 ....: t.sphere((i,i^2 - 0.5,i^3), 0.1, 't%s'%(k%3)) ....: t.cylinder((0,0,0), (0,0,1), 0.05,'t1') sage: t.show()
Many random spheres::
sage: t = Tachyon(xres=512,yres=512, camera_center=(2,0.5,0.5), look_at=(0.5,0.5,0.5), raydepth=4) sage: t.light((4,3,2), 0.2, (1,1,1)) sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0)) sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0)) sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0)) sage: k=0 sage: for i in range(100): ....: k += 1 ....: t.sphere((random(),random(), random()), random()/10, 't%s'%(k%3)) sage: t.show()
Points on an elliptic curve, their height indicated by their height above the axis::
sage: t = Tachyon(camera_center=(5,2,2), look_at=(0,1,0)) sage: t.light((10,3,2), 0.2, (1,1,1)) sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1,0,0)) sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(0,1,0)) sage: t.texture('t2', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(0,0,1)) sage: E = EllipticCurve('37a') sage: P = E([0,0]) sage: Q = P sage: n = 100 sage: for i in range(n): # increase 20 for a better plot ....: Q = Q + P ....: t.sphere((Q[1], Q[0], ZZ(i)/n), 0.1, 't%s'%(i%3)) sage: t.show()
A beautiful picture of rational points on a rank 1 elliptic curve.
::
sage: t = Tachyon(xres=1000, yres=800, camera_center=(2,7,4), look_at=(2,0,0), raydepth=4) sage: t.light((10,3,2), 1, (1,1,1)) sage: t.light((10,-3,2), 1, (1,1,1)) sage: t.texture('black', color=(0,0,0)) sage: t.texture('red', color=(1,0,0)) sage: t.texture('grey', color=(.9,.9,.9)) sage: t.plane((0,0,0),(0,0,1),'grey') sage: t.cylinder((0,0,0),(1,0,0),.01,'black') sage: t.cylinder((0,0,0),(0,1,0),.01,'black') sage: E = EllipticCurve('37a') sage: P = E([0,0]) sage: Q = P sage: n = 100 sage: for i in range(n): ....: Q = Q + P ....: c = i/n + .1 ....: t.texture('r%s'%i,color=(float(i/n),0,0)) ....: t.sphere((Q[0], -Q[1], .01), .04, 'r%s'%i) sage: t.show() # long time, e.g., 10-20 seconds
A beautiful spiral.
::
sage: t = Tachyon(xres=800,yres=800, camera_center=(2,5,2), look_at=(2.5,0,0)) sage: t.light((0,0,100), 1, (1,1,1)) sage: t.texture('r', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1,0,0)) sage: for i in srange(0,50,0.1): ....: t.sphere((i/10,sin(i),cos(i)), 0.05, 'r') sage: t.texture('white', color=(1,1,1), opacity=1, specular=1, diffuse=1) sage: t.plane((0,0,-100), (0,0,-100), 'white') sage: t.show()
If the optional parameter ``viewdir`` is not set, the camera center should not coincide with the point which is looked at (see :trac:`7232`)::
sage: t = Tachyon(xres=80,yres=80, camera_center=(2,5,2), look_at=(2,5,2)) Traceback (most recent call last): ... ValueError: camera_center and look_at coincide
Use of a fisheye lens perspective. ::
sage: T = Tachyon(xres=800, yres=600, camera_center=(-1.5,-1.5,.3), projection='fisheye', frustum=(-1.0, 1.0, -1.0, 1.0)) sage: T.texture('t1', color=(0,0,1)) sage: cedges = [[[1, 1, 1], [-1, 1, 1]], [[1, 1, 1], [1, -1, 1]], ....: [[1, 1, 1], [1, 1, -1]], [[-1, 1, 1], [-1, -1, 1]], [[-1, 1, 1], ....: [-1, 1, -1]], [[1, -1, 1], [-1, -1, 1]], [[1, -1, 1], ....: [1, -1, -1]], ....: [[-1, -1, 1], [-1, -1, -1]], [[1, 1, -1], [-1, 1, -1]], ....: [[1, 1, -1], [1, -1, -1]], [[-1, 1, -1], [-1, -1, -1]], ....: [[1, -1, -1], [-1, -1, -1]]] sage: for ed in cedges: ....: T.fcylinder(ed[0], ed[1], .05, 't1') sage: T.light((-4,-4,4), .1, (1,1,1)) sage: T.show()
Use of the ``projection='perspective_dof'`` option. This may not be implemented correctly. ::
sage: T = Tachyon(xres=800,antialiasing=4, raydepth=10, projection='perspective_dof', focallength='1.0', aperture='.0025') sage: T.light((0,5,7), 1.0, (1,1,1)) sage: T.texture('t1', opacity=1, specular=.3) sage: T.texture('t2', opacity=1, specular=.3, color=(0,0,1)) sage: T.texture('t3', opacity=1, specular=1, color=(1,.8,1), diffuse=0.2) sage: T.plane((0,0,-1), (0,0,1), 't3') sage: ttlist = ['t1', 't2'] sage: tt = 't1' sage: T.cylinder((0,0,.1), (1,1/3,0), .05, 't3') sage: for q in srange(-3, 100, .15): ....: if tt == 't1': ....: tt = 't2' ....: else: ....: tt = 't1' ....: T.sphere((q, q/3+.3*sin(3*q), .1+.3*cos(3*q)), .1, tt) sage: T.show()
TESTS::
sage: hash(Tachyon()) # random 140658972348064 """ def __init__(self, xres=350, yres=350, zoom=1.0, antialiasing=False, aspectratio=1.0, raydepth=8, camera_center=(-3, 0, 0), updir=(0, 0, 1), look_at=(0, 0, 0), viewdir=None, projection='PERSPECTIVE', focallength='', aperture='', frustum=''): r""" Create an instance of the Tachyon class.
EXAMPLES::
sage: t = Tachyon() sage: t._xres 350 """ for i in range(3)] else: else:
def save_image(self, filename=None, *args, **kwds): r""" Save an image representation of ``self``.
The image type is determined by the extension of the filename. For example, this could be ``.png``, ``.jpg``, ``.gif``, ``.pdf``, ``.svg``. Currently this is implemented by calling the :meth:`save` method of self, passing along all arguments and keywords.
.. NOTE::
Not all image types are necessarily implemented for all graphics types. See :meth:`save` for more details.
EXAMPLES::
sage: q = Tachyon() sage: q.light((1,1,11), 1,(1,1,1)) sage: q.texture('s') sage: q.sphere((0,-1,1),1,'s') sage: tempname = tmp_filename() sage: q.save_image(tempname)
TESTS:
:meth:`save_image` is used for generating animations::
sage: def tw_cubic(t): ....: q = Tachyon() ....: q.light((1,1,11), 1,(1,1,1)) ....: q.texture('s') ....: for i in srange(-1,t,0.05): ....: q.sphere((i,i^2-0.5,i^3), 0.1, 's') ....: return q
sage: a = animate([tw_cubic(t) for t in srange(-1,1,.3)]) sage: a # optional -- ImageMagick Animation with 7 frames sage: a.show() # optional -- ImageMagick """
def save(self, filename='sage.png', verbose=None, extra_opts=''): r""" Save rendering of the tachyon scene
INPUT:
- ``filename`` - (default: 'sage.png') output filename; the extension of the filename determines the type. Supported types include:
- ``tga`` - 24-bit (uncompressed)
- ``bmp`` - 24-bit Windows BMP (uncompressed)
- ``ppm`` - 24-bit PPM (uncompressed)
- ``rgb`` - 24-bit SGI RGB (uncompressed)
- ``png`` - 24-bit PNG (compressed, lossless)
- ``verbose`` - integer (default: None); if no verbosity setting is supplied, the verbosity level set by sage.misc.misc.set_verbose is used.
- ``0`` - silent
- ``1`` - some output
- ``2`` - very verbose output
- ``extra_opts`` - passed directly to tachyon command line. Use tachyon_rt.usage() to see some of the possibilities.
EXAMPLES::
sage: q = Tachyon() sage: q.light((1,1,11), 1,(1,1,1)) sage: q.texture('s') sage: q.sphere((0,0,0),1,'s') sage: tempname = tmp_filename() sage: q.save(tempname) """
def _rich_repr_(self, display_manager, **kwds): """ Rich Output Magic Method
See :mod:`sage.repl.rich_output` for details.
EXAMPLES::
sage: q = Tachyon() sage: q.light((1,1,11), 1,(1,1,1)) sage: q.texture('s') sage: q.sphere((0,0,0),1,'s') sage: from sage.repl.rich_output import get_display_manager sage: dm = get_display_manager() sage: q._rich_repr_(dm) OutputImagePng container """
def show(self, **kwds): r""" Create a PNG file of the scene.
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.
OUTPUT:
This method does not return anything. Use :meth:`save` if you want to save the figure as an image.
EXAMPLES:
This example demonstrates how the global Sage verbosity setting is used if none is supplied. Firstly, using a global verbosity setting of 0 means no extra technical information is displayed, and we are simply shown the plot.
::
sage: h = Tachyon(xres=512,yres=512, camera_center=(4,-4,3),viewdir=(-4,4,-3), raydepth=4) sage: h.light((4.4,-4.4,4.4), 0.2, (1,1,1)) sage: def f(x,y): return float(sin(x*y)) sage: h.texture('t0', ambient=0.1, diffuse=0.9, specular=0.1, opacity=1.0, color=(1.0,0,0)) sage: h.plot(f,(-4,4),(-4,4),"t0",max_depth=5,initial_depth=3, num_colors=60) # increase min_depth for better picture sage: set_verbose(0) sage: h.show()
This second example, using a "medium" global verbosity setting of 1, displays some extra technical information then displays our graph.
::
sage: s = Tachyon(xres=512,yres=512, camera_center=(4,-4,3),viewdir=(-4,4,-3), raydepth=4) sage: s.light((4.4,-4.4,4.4), 0.2, (1,1,1)) sage: def f(x,y): return float(sin(x*y)) sage: s.texture('t0', ambient=0.1, diffuse=0.9, specular=0.1, opacity=1.0, color=(1.0,0,0)) sage: s.plot(f,(-4,4),(-4,4),"t0",max_depth=5,initial_depth=3, num_colors=60) # increase min_depth for better picture sage: set_verbose(1) sage: s.show() tachyon ... Scene contains 2713 objects. ...
The last example shows how you can override the global Sage verbosity setting, my supplying a setting level as an argument. In this case we chose the highest verbosity setting level, 2, so much more extra technical information is shown, along with the plot.
::
sage: set_verbose(0) sage: d = Tachyon(xres=512,yres=512, camera_center=(4,-4,3),viewdir=(-4,4,-3), raydepth=4) sage: d.light((4.4,-4.4,4.4), 0.2, (1,1,1)) sage: def f(x,y): return float(sin(x*y)) sage: d.texture('t0', ambient=0.1, diffuse=0.9, specular=0.1, opacity=1.0, color=(1.0,0,0)) sage: d.plot(f,(-4,4),(-4,4),"t0",max_depth=5,initial_depth=3, num_colors=60) # increase min_depth for better picture sage: get_verbose() 0 sage: d.show(verbose=2) tachyon ... Scene contains 2713 objects. ... Scene contains 1 non-gridded objects ... """
def _res(self): r""" An internal function that writes the tachyon string for the resolution (x and y size of the image).
EXAMPLES::
sage: t = Tachyon(xres = 300, yres = 700) sage: t._res() '\nresolution 300 700\n' """
def _camera(self): r""" An internal function that writes the tachyon string for the camera and other rendering information (ray depth, antialiasing).
EXAMPLES::
sage: t = Tachyon(raydepth = 16, zoom = 2, antialiasing = True) sage: t._camera().split()[3:10] ['zoom', '2.0', 'aspectratio', '1.0', 'antialiasing', '1', 'raydepth'] """ camera projection %s"""%(tostr(self._projection)) focallength %s"""%(float(self._focallength)) aperture %s"""%(float(self._aperture)) zoom %s aspectratio %s antialiasing %s raydepth %s center %s viewdir %s updir %s"""%(float(self._zoom), float(self._aspectratio), int(self._antialiasing), int(self._raydepth), tostr(self._camera_center), tostr(self._viewdir), tostr(self._updir)) frustum %s"""%(tostr(self._frustum)) end_camera"""
def str(self): r""" Return the complete tachyon scene file as a string.
EXAMPLES::
sage: t = Tachyon(xres=500,yres=500, camera_center=(2,0,0)) sage: t.light((4,3,2), 0.2, (1,1,1)) sage: t.texture('t2', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1,0,0)) sage: t.texture('t3', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(0,1,0)) sage: t.texture('t4', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(0,0,1)) sage: t.sphere((0,0.5,0), 0.2, 't2') sage: t.sphere((0.5,0,0), 0.2, 't3') sage: t.sphere((0,0,0.5), 0.2, 't4') sage: 'PLASTIC' in t.str() True """ begin_scene %s %s %s end_scene"""%( self._res(), self._camera(), '\n'.join([x.str() for x in self._objects]) )
def light(self, center, radius, color): r""" Create a light source of the given center, radius, and color.
EXAMPLES::
sage: q = Tachyon() sage: q.light((1,1,1),1.0,(.2,0,.8)) sage: q.str().split('\n')[17] ' light center 1.0 1.0 1.0 ' """
def texfunc(self, type=0, center=(0,0,0), rotate=(0,0,0), scale=(1,1,1), imagefile=''): r""" INPUT:
- ``type`` - (default: 0)
0. No special texture, plain shading 1. 3D checkerboard function, like a rubik's cube 2. Grit Texture, randomized surface color 3. 3D marble texture, uses object's base color 4. 3D wood texture, light and dark brown, not very good yet 5. 3D gradient noise function (can't remember what it looks like) 6. Don't remember 7. Cylindrical Image Map, requires ppm filename (with path) 8. Spherical Image Map, requires ppm filename (with path) 9. Planar Image Map, requires ppm filename (with path)
- ``center`` - (default: (0,0,0)) - ``rotate`` - (default: (0,0,0)) - ``scale`` - (default: (1,1,1))
EXAMPLES: We draw an infinite checkerboard::
sage: t = Tachyon(camera_center=(2,7,4), look_at=(2,0,0)) sage: t.texture('black', color=(0,0,0), texfunc=1) sage: t.plane((0,0,0),(0,0,1),'black') sage: t.show() """ raise ValueError("type must be an integer between 0 and 9")
def texture(self, name, ambient=0.2, diffuse=0.8, specular=0.0, opacity=1.0, color=(1.0,0.0, 0.5), texfunc=0, phong=0, phongsize=.5, phongtype="PLASTIC", imagefile=''): r""" INPUT:
- ``name`` - string; the name of the texture (to be used later)
- ``ambient`` - (default: 0.2)
- ``diffuse`` - (default: 0.8)
- ``specular`` - (default: 0.0)
- ``opacity`` - (default: 1.0)
- ``color`` - (default: (1.0,0.0,0.5))
- ``texfunc`` - (default: 0); a texture function; this is either the output of self.texfunc, or a number between 0 and 9, inclusive. See the docs for self.texfunc.
- ``phong`` - (default: 0)
- ``phongsize`` - (default: 0.5)
- ``phongtype`` - (default: "PLASTIC")
EXAMPLES:
We draw a scene with 4 spheres that illustrates various uses of the texture command::
sage: t = Tachyon(camera_center=(2,5,4), look_at=(2,0,0), raydepth=6) sage: t.light((10,3,4), 1, (1,1,1)) sage: t.texture('mirror', ambient=0.05, diffuse=0.05, specular=.9, opacity=0.9, color=(.8,.8,.8)) sage: t.texture('grey', color=(.8,.8,.8), texfunc=3) sage: t.plane((0,0,0),(0,0,1),'grey') sage: t.sphere((4,-1,1), 1, 'mirror') sage: t.sphere((0,-1,1), 1, 'mirror') sage: t.sphere((2,-1,1), 0.5, 'mirror') sage: t.sphere((2,1,1), 0.5, 'mirror') sage: show(t) # known bug (trac #7232) """ specular, opacity, color, texfunc, phong, phongsize, phongtype, imagefile=imagefile))
def texture_recolor(self, name, colors): r""" Recolor default textures.
EXAMPLES::
sage: t = Tachyon() sage: t.texture('s') sage: q = t.texture_recolor('s',[(0,0,1)]) sage: t._objects[1]._color (0.0, 0.0, 1.0) """
base_tex = Texture(name)
def sphere(self, center, radius, texture): r""" Create the scene information for a sphere with the given center, radius, and texture.
EXAMPLES::
sage: t = Tachyon() sage: t.texture('sphere_texture') sage: t.sphere((1,2,3), .1, 'sphere_texture') sage: t._objects[1].str() '\n sphere center 1.0 2.0 3.0 rad 0.1 sphere_texture\n ' """
def ring(self, center, normal, inner, outer, texture): r""" Creates the scene information for a ring with the given parameters.
EXAMPLES::
sage: t = Tachyon() sage: t.ring([0,0,0], [0,0,1], 1.0, 2.0, 's') sage: t._objects[0]._center (0.0, 0.0, 0.0) """
def cylinder(self, center, axis, radius, texture): r""" Creates the scene information for a infinite cylinder with the given center, axis direction, radius, and texture.
EXAMPLES::
sage: t = Tachyon() sage: t.texture('c') sage: t.cylinder((0,0,0),(-1,-1,-1),.1,'c') """
def plane(self, center, normal, texture): r""" Creates an infinite plane with the given center and normal.
TESTS::
sage: t = Tachyon() sage: t.plane((0,0,0),(1,1,1),'s') sage: plane_pos = t.str().index('plane') sage: t.str()[plane_pos:plane_pos+42] 'plane center 0.0 0.0 0.0 normal 1.0 1.0' """
def axis_aligned_box(self, min_p, max_p, texture): r""" Creates an axis-aligned box with minimal point ``min_p`` and maximum point ``max_p``.
EXAMPLES::
sage: t = Tachyon() sage: t.axis_aligned_box((0,0,0),(2,2,2),'s') """
def fcylinder(self, base, apex, radius, texture): r""" Finite cylinders are almost the same as infinite ones, but the center and length of the axis determine the extents of the cylinder. The finite cylinder is also really a shell, it doesn't have any caps. If you need to close off the ends of the cylinder, use two ring objects, with the inner radius set to 0.0 and the normal set to be the axis of the cylinder. Finite cylinders are built this way to enhance speed.
EXAMPLES::
sage: t = Tachyon() sage: t.fcylinder((1,1,1),(1,2,3),.01,'s') sage: len(t.str()) 451 """
def triangle(self, vertex_1, vertex_2, vertex_3, texture): r""" Creates a triangle with the given vertices and texture.
EXAMPLES::
sage: t = Tachyon() sage: t.texture('s') sage: t.triangle([1,2,3],[4,5,6],[7,8,10],'s') sage: t._objects[1].get_vertices() ([1, 2, 3], [4, 5, 6], [7, 8, 10])
"""
def smooth_triangle(self, vertex_1, vertex_2, vertex_3, normal_1, normal_2, normal_3, texture): r""" Creates a triangle along with a normal vector for smoothing.
EXAMPLES::
sage: t = Tachyon() sage: t.light((1,1,1),.1,(1,1,1)) sage: t.texture('s') sage: t.smooth_triangle([0,0,0],[0,0,1],[0,1,0],[0,1,1],[-1,1,2],[3,0,0],'s') sage: t._objects[2].get_vertices() ([0, 0, 0], [0, 0, 1], [0, 1, 0]) sage: t._objects[2].get_normals() ([0, 1, 1], [-1, 1, 2], [3, 0, 0]) """
def fractal_landscape(self, res, scale, center, texture): r""" Axis-aligned fractal landscape. Not very useful at the moment.
EXAMPLES::
sage: t = Tachyon() sage: t.texture('s') sage: t.fractal_landscape([30,30],[80,80],[0,0,0],'s') sage: len(t._objects) 2 """
def plot(self, f, xmin_xmax, ymin_ymax, texture, grad_f=None, max_bend=.7, max_depth=5, initial_depth=3, num_colors=None): r""" INPUT:
- ``f`` - Function of two variables, which returns a float (or coercible to a float) (xmin,xmax)
- ``(ymin,ymax)`` - defines the rectangle to plot over texture: Name of texture to be used Optional arguments:
- ``grad_f`` - gradient function. If specified, smooth triangles will be used.
- ``max_bend`` - Cosine of the threshold angle between triangles used to determine whether or not to recurse after the minimum depth
- ``max_depth`` - maximum recursion depth. Maximum triangles plotted = `2^{2*max_depth}`
- ``initial_depth`` - minimum recursion depth. No error-tolerance checking is performed below this depth. Minimum triangles plotted: `2^{2*min_depth}`
- ``num_colors`` - Number of rainbow bands to color the plot with. Texture supplied will be cloned (with different colors) using the texture_recolor method of the Tachyon object.
Plots a function by constructing a mesh with nonstandard sampling density without gaps. At very high resolutions (depths 10) it becomes very slow. Cython may help. Complexity is approx. `O(2^{2*maxdepth})`. This algorithm has been optimized for speed, not memory - values from f(x,y) are recycled rather than calling the function multiple times. At high recursion depth, this may cause problems for some machines.
Flat Triangles::
sage: t = Tachyon(xres=512,yres=512, camera_center=(4,-4,3),viewdir=(-4,4,-3), raydepth=4) sage: t.light((4.4,-4.4,4.4), 0.2, (1,1,1)) sage: def f(x,y): return float(sin(x*y)) sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.1, opacity=1.0, color=(1.0,0,0)) sage: t.plot(f,(-4,4),(-4,4),"t0",max_depth=5,initial_depth=3, num_colors=60) # increase min_depth for better picture sage: t.show(verbose=1) tachyon ... Scene contains 2713 objects. ...
Plotting with Smooth Triangles (requires explicit gradient function)::
sage: t = Tachyon(xres=512,yres=512, camera_center=(4,-4,3),viewdir=(-4,4,-3), raydepth=4) sage: t.light((4.4,-4.4,4.4), 0.2, (1,1,1)) sage: def f(x,y): return float(sin(x*y)) sage: def g(x,y): return ( float(y*cos(x*y)), float(x*cos(x*y)), 1 ) sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.1, opacity=1.0, color=(1.0,0,0)) sage: t.plot(f,(-4,4),(-4,4),"t0",max_depth=5,initial_depth=3, grad_f = g) # increase min_depth for better picture sage: t.show(verbose=1) tachyon ... Scene contains 2713 objects. ...
Preconditions: f is a scalar function of two variables, grad_f is None or a triple-valued function of two variables, min_x != max_x, min_y != max_y
::
sage: f = lambda x,y: x*y sage: t = Tachyon() sage: t.plot(f,(2.,2.),(-2.,2.),'') Traceback (most recent call last): ... ValueError: Plot rectangle is really a line. Make sure min_x != max_x and min_y != max_y. """ min_depth=initial_depth, max_depth=max_depth, max_bend=max_bend, num_colors = num_colors)
def parametric_plot(self, f, t_0, t_f, tex, r=.1, cylinders = True, min_depth=4, max_depth=8, e_rel = .01, e_abs = .01): r""" Plots a space curve as a series of spheres and finite cylinders. Example (twisted cubic) ::
sage: f = lambda t: (t,t^2,t^3) sage: t = Tachyon(camera_center=(5,0,4)) sage: t.texture('t') sage: t.light((-20,-20,40), 0.2, (1,1,1)) sage: t.parametric_plot(f,-5,5,'t',min_depth=6) sage: t.show(verbose=1) tachyon ... Scene contains 514 objects. ... """ ParametricPlot( f, t_0, t_f, tex, r=r, cylinders=cylinders, min_depth=min_depth, max_depth=max_depth, e_rel=.01,e_abs=.01 ) )
class Light(object): r""" Represents lighting objects.
EXAMPLES::
sage: from sage.plot.plot3d.tachyon import Light sage: q = Light((1,1,1), 1, (1,1,1)) sage: q._center (1.0, 1.0, 1.0) """ def __init__(self, center, radius, color): r""" Stores the center, radius and color.
EXAMPLES::
sage: from sage.plot.plot3d.tachyon import Light sage: q = Light((1,1,1), 1, (1,1,1)) sage: q._color (1.0, 1.0, 1.0) """
def str(self): r""" Returns the tachyon string defining the light source.
EXAMPLES::
sage: from sage.plot.plot3d.tachyon import Light sage: q = Light((1,1,1), 1, (1,1,1)) sage: q._radius 1.0 """ light center %s rad %s color %s """%(tostr(self._center), self._radius, tostr(self._color))
class Texfunc(object):
def __init__(self, ttype=0, center=(0,0,0), rotate=(0,0,0), scale=(1,1,1), imagefile=''): r""" Creates a texture function.
EXAMPLES::
sage: from sage.plot.plot3d.tachyon import Texfunc sage: t = Texfunc() sage: t._ttype 0 """
def str(self): r""" Returns the scene string for this texture function.
EXAMPLES::
sage: from sage.plot.plot3d.tachyon import Texfunc sage: t = Texfunc() sage: t.str() '0' """ self._ttype, tostr(self._center), tostr(self._rotate), tostr(self._scale)) elif self._ttype < 9: return r"""%d %s center %s rotate %s scale %s"""%( self._ttype, self._imagefile, tostr(self._center), tostr(self._rotate), tostr(self._scale)) elif self._ttype == 9: return r"""%d %s center %s rotate %s scale %s uaxis 1.0 0.0 0.0 vaxis 0.0 1.0 0.0"""%( self._ttype, self._imagefile, tostr(self._center), tostr(self._rotate), tostr(self._scale)) else: raise ValueError
class Texture(object):
def __init__(self, name, ambient=0.2, diffuse=0.8, specular=0.0, opacity=1.0, color=(1.0,0.0, 0.5), texfunc=0, phong=0, phongsize=0, phongtype="PLASTIC", imagefile=''): r""" Stores texture information.
EXAMPLES::
sage: from sage.plot.plot3d.tachyon import Texture sage: t = Texture('w') sage: t.str().split()[2:6] ['ambient', '0.2', 'diffuse', '0.8'] """
def recolor(self, name, color): r""" Returns a texture with the new given color.
EXAMPLES::
sage: from sage.plot.plot3d.tachyon import Texture sage: t2 = Texture('w') sage: t2w = t2.recolor('w2', (.1,.2,.3)) sage: t2ws = t2w.str() sage: color_index = t2ws.find('color') sage: t2ws[color_index:color_index+20] 'color 0.1 0.2 0.3 ' """ color, self._texfunc, self._phong, self._phongsize, self._phongtype, self._imagefile)
def str(self): r""" Returns the scene string for this texture.
EXAMPLES::
sage: from sage.plot.plot3d.tachyon import Texture sage: t = Texture('w') sage: t.str().split()[2:6] ['ambient', '0.2', 'diffuse', '0.8'] """ texdef %s ambient %s diffuse %s specular %s opacity %s phong %s %s phong_size %s color %s texfunc %s """%(self._name, self._ambient, self._diffuse, self._specular, self._opacity, self._phongtype, self._phong, self._phongsize, tostr(self._color), self._texfunc)
class Sphere(object): r""" A class for creating spheres in tachyon. """ def __init__(self, center, radius, texture): r""" Stores the center, radius, and texture information in a class.
EXAMPLES::
sage: t = Tachyon() sage: from sage.plot.plot3d.tachyon import Sphere sage: t.texture('r', color=(.8,0,0), ambient=.1) sage: s = Sphere((1,1,1), 1, 'r') sage: s._radius 1.0 """
def str(self): r""" Returns the scene string for the sphere.
EXAMPLES::
sage: t = Tachyon() sage: from sage.plot.plot3d.tachyon import Sphere sage: t.texture('r', color=(.8,0,0), ambient = .1) sage: s = Sphere((1,1,1), 1, 'r') sage: s.str() '\n sphere center 1.0 1.0 1.0 rad 1.0 r\n ' """ sphere center %s rad %s %s """%(tostr(self._center), self._radius, self._texture)
class Ring(object): r""" An annulus of zero thickness. """ def __init__(self, center, normal, inner, outer, texture): r""" Creates a ring with the given center, normal, inner radius, outer radius, and texture.
EXAMPLES::
sage: from sage.plot.plot3d.tachyon import Ring sage: r = Ring((1,1,1), (1,1,0), 1.0, 2.0, 's') sage: r._center (1.0, 1.0, 1.0) """
def str(self): r""" Returns the scene string of the ring.
EXAMPLES::
sage: from sage.plot.plot3d.tachyon import Ring sage: r = Ring((0,0,0), (1,1,0), 1.0, 2.0, 's') sage: r.str() '\n ring center 0.0 0.0 0.0 normal 1.0 1.0 0.0 inner 1.0 outer 2.0 s\n ' """ ring center %s normal %s inner %s outer %s %s """%(tostr(self._center), tostr(self._normal), self._inner, self._outer, self._texture)
class FractalLandscape(object): r""" Axis-aligned fractal landscape. Does not seem very useful at the moment, but perhaps will be improved in the future. """ def __init__(self, res, scale, center, texture): r""" Creates a fractal landscape in tachyon.
EXAMPLES::
sage: from sage.plot.plot3d.tachyon import FractalLandscape sage: fl = FractalLandscape([20,20],[30,30],[1,2,3],'s') sage: fl._center (1.0, 2.0, 3.0) """
def str(self): r""" Returns the scene string of the fractal landscape.
EXAMPLES::
sage: from sage.plot.plot3d.tachyon import FractalLandscape sage: fl = FractalLandscape([20,20],[30,30],[1,2,3],'s') sage: fl.str() '\n scape res 20 20 scale 30 30 center 1.0 2.0 3.0 s\n ' """ scape res %s scale %s center %s %s """%(tostr(self._res, 2, int), tostr(self._scale, 2, int), tostr(self._center), self._texture)
class Cylinder(object): r""" An infinite cylinder. """ def __init__(self, center, axis, radius, texture): r""" Creates a cylinder with the given parameters.
EXAMPLES::
sage: t = Tachyon() sage: from sage.plot.plot3d.tachyon import Cylinder sage: c = Cylinder((0,0,0),(1,1,1),.1,'s') sage: c.str() '\n cylinder center 0.0 0.0 0.0 axis 1.0 1.0 1.0 rad 0.1 s\n ' """
def str(self): r""" Returns the scene string of the cylinder.
EXAMPLES::
sage: t = Tachyon() sage: from sage.plot.plot3d.tachyon import Cylinder sage: c = Cylinder((0,0,0),(1,1,1),.1,'s') sage: c.str() '\n cylinder center 0.0 0.0 0.0 axis 1.0 1.0 1.0 rad 0.1 s\n ' """ cylinder center %s axis %s rad %s %s """%(tostr(self._center), tostr(self._axis), self._radius, self._texture)
class Plane(object): r""" An infinite plane. """ def __init__(self, center, normal, texture): r""" Creates the plane object.
EXAMPLES::
sage: from sage.plot.plot3d.tachyon import Plane sage: p = Plane((1,2,3), (1,2,4), 's') sage: p.str() '\n plane center 1.0 2.0 3.0 normal 1.0 2.0 4.0 s\n ' """
def str(self): r""" Returns the scene string of the plane.
EXAMPLES::
sage: from sage.plot.plot3d.tachyon import Plane sage: p = Plane((1,2,3),(1,2,4),'s') sage: p.str() '\n plane center 1.0 2.0 3.0 normal 1.0 2.0 4.0 s\n ' """ plane center %s normal %s %s """%(tostr(self._center), tostr(self._normal), self._texture)
class FCylinder(object): r""" A finite cylinder. """ def __init__(self, base, apex, radius, texture): r""" Creates a finite cylinder object.
EXAMPLES::
sage: from sage.plot.plot3d.tachyon import FCylinder sage: fc = FCylinder((0,0,0),(1,1,1),.1,'s') sage: fc.str() '\n fcylinder base 0.0 0.0 0.0 apex 1.0 1.0 1.0 rad 0.1 s\n ' """
def str(self): r""" Returns the scene string of the finite cylinder.
EXAMPLES::
sage: from sage.plot.plot3d.tachyon import FCylinder sage: fc = FCylinder((0,0,0),(1,1,1),.1,'s') sage: fc.str() '\n fcylinder base 0.0 0.0 0.0 apex 1.0 1.0 1.0 rad 0.1 s\n ' """ fcylinder base %s apex %s rad %s %s """%(tostr(self._center), tostr(self._axis), self._radius, self._texture)
class Axis_aligned_box(object): r""" Box with axis-aligned edges with the given min and max coordinates. """ def __init__(self, min_p, max_p, texture): r""" Creates the axis-aligned box object.
EXAMPLES::
sage: from sage.plot.plot3d.tachyon import Axis_aligned_box sage: aab = Axis_aligned_box((0,0,0),(1,1,1),'s') sage: aab.str() '\n box min 0.0 0.0 0.0 max 1.0 1.0 1.0 s\n ' """
def str(self): r""" Returns the scene string of the axis-aligned box.
EXAMPLES::
sage: from sage.plot.plot3d.tachyon import Axis_aligned_box sage: aab = Axis_aligned_box((0,0,0),(1,1,1),'s') sage: aab.str() '\n box min 0.0 0.0 0.0 max 1.0 1.0 1.0 s\n ' """ box min %s max %s %s """%(tostr(self._min_p), tostr(self._max_p), self._texture)
class TachyonTriangle(Triangle): r""" Basic triangle class. """ def str(self): r""" Returns the scene string for a triangle.
EXAMPLES::
sage: from sage.plot.plot3d.tachyon import TachyonTriangle sage: t = TachyonTriangle([-1,-1,-1],[0,0,0],[1,2,3]) sage: t.str() '\n TRI V0 -1.0 -1.0 -1.0 V1 0.0 0.0 0.0 V2 1.0 2.0 3.0 \n 0\n ' """ TRI V0 %s V1 %s V2 %s %s """%(tostr(self._a), tostr(self._b), tostr(self._c), self._color)
class TachyonSmoothTriangle(SmoothTriangle): r""" A triangle along with a normal vector, which is used for smoothing. """ def str(self): r""" Return the scene string for a smoothed triangle.
EXAMPLES::
sage: from sage.plot.plot3d.tachyon import TachyonSmoothTriangle sage: t = TachyonSmoothTriangle([-1,-1,-1],[0,0,0],[1,2,3],[1,0,0],[0,1,0],[0,0,1]) sage: t.str() '\n STRI V0 ... 1.0 0.0 0.0 N1 0.0 1.0 0.0 N2 0.0 0.0 1.0 \n 0\n ' """ STRI V0 %s V1 %s V2 %s N0 %s N1 %s N2 %s %s """%(tostr(self._a), tostr(self._b), tostr(self._c), tostr(self._da), tostr(self._db), tostr(self._dc), self._color)
class TachyonTriangleFactory(TriangleFactory): r""" A class to produce triangles of various rendering types. """ def __init__(self, tach, tex): r""" Initializes with tachyon instance and texture.
EXAMPLES::
sage: from sage.plot.plot3d.tachyon import TachyonTriangleFactory sage: t = Tachyon() sage: t.texture('s') sage: ttf = TachyonTriangleFactory(t, 's') sage: ttf._texture 's' """
def triangle(self,a,b,c,color=None): r""" Creates a TachyonTriangle with vertices a, b, and c.
EXAMPLES::
sage: from sage.plot.plot3d.tachyon import TachyonTriangleFactory sage: t = Tachyon() sage: t.texture('s') sage: ttf = TachyonTriangleFactory(t, 's') sage: ttft = ttf.triangle([1,2,3],[3,2,1],[0,2,1]) sage: ttft.str() '\n TRI V0 1.0 2.0 3.0 V1 3.0 2.0 1.0 V2 0.0 2.0 1.0 \n s\n ' """ else: return TachyonTriangle(a,b,c,color)
def smooth_triangle(self,a,b,c,da,db,dc,color=None): r""" Creates a TachyonSmoothTriangle.
EXAMPLES::
sage: from sage.plot.plot3d.tachyon import TachyonTriangleFactory sage: t = Tachyon() sage: t.texture('s') sage: ttf = TachyonTriangleFactory(t, 's') sage: ttfst = ttf.smooth_triangle([0,0,0],[1,0,0],[0,0,1],[1,1,1],[1,2,3],[-1,-1,2]) sage: ttfst.str() '\n STRI V0 0.0 0.0 0.0 ...' """ else: return TachyonSmoothTriangle(a,b,c,da,db,dc,color)
def get_colors(self, list): r""" Returns a list of color labels.
EXAMPLES::
sage: from sage.plot.plot3d.tachyon import TachyonTriangleFactory sage: t = Tachyon() sage: t.texture('s') sage: ttf = TachyonTriangleFactory(t, 's') sage: ttf.get_colors([(1,1,1)]) ['SAGETEX1_0'] """
class ParametricPlot(object): r""" Parametric plotting routines. """ def str(self): r""" Returns the tachyon string representation of the parameterized curve.
EXAMPLES::
sage: from sage.plot.plot3d.tachyon import ParametricPlot sage: t = var('t') sage: f = lambda t: (t,t^2,t^3) sage: q = ParametricPlot(f,0,1,'s') sage: q.str()[9:69] 'sphere center 0.0 0.0 0.0 rad 0.1 s\n \n fcyli' """
def __init__(self, f, t_0, t_f, tex, r=.1, cylinders = True, min_depth=4, max_depth=8, e_rel = .01, e_abs = .01): r""" Creates the parametric plotting class.
EXAMPLES::
sage: from sage.plot.plot3d.tachyon import ParametricPlot sage: t = var('t') sage: f = lambda t: (t,t^2,t^3) sage: q = ParametricPlot(f,0,1,'s') sage: q._e_rel 0.01 """
def _plot_step(self, depth, t_0,t_f,f_0,f_f): r""" Recursively subdivides interval, eventually plotting with cylinders and spheres.
EXAMPLES::
sage: from sage.plot.plot3d.tachyon import ParametricPlot sage: t = var('t') sage: f = lambda t: (t,t^2,t^3) sage: q = ParametricPlot(f,0,1,'s') sage: q._plot_step(8,0,1,[0,0,0],[1,1,1]) sage: len(q._objects) 515 """ else: new_depth = self._max_depth
else:
def tol(self, est, val): r""" Check relative, then absolute tolerance. If both fail, return False. This is a zero-safe error checker.
EXAMPLES::
sage: from sage.plot.plot3d.tachyon import ParametricPlot sage: t = var('t') sage: f = lambda t: (t,t^2,t^3) sage: q = ParametricPlot(f,0,1,'s') sage: q.tol([0,0,0],[1,0,0]) False sage: q.tol([0,0,0],[.0001,0,0]) True """
return True
def tostr(s, length=3, out_type=float): r""" Converts vector information to a space-separated string.
EXAMPLES::
sage: from sage.plot.plot3d.tachyon import tostr sage: tostr((1,1,1)) ' 1.0 1.0 1.0 ' sage: tostr('2 3 2') '2 3 2' """
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