Coverage for local/lib/python2.7/site-packages/sage/plot/plot3d/shapes2.py : 85%

r""" 

Classes for Lines, Frames, Rulers, Spheres, Points, Dots, and Text 

AUTHORS: 

- William Stein (2007-12): initial version 

- William Stein and Robert Bradshaw (2008-01): Many improvements 

""" 

#***************************************************************************** 

# Copyright (C) 2007 William Stein <wstein@gmail.com> 

# Copyright (C) 2008 Robert Bradshaw <robertwb@math.washington.edu> 

# 

# Distributed under the terms of the GNU General Public License (GPL) 

# 

# This code is distributed in the hope that it will be useful, 

# but WITHOUT ANY WARRANTY; without even the implied warranty of 

# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU 

# General Public License for more details. 

# 

# The full text of the GPL is available at: 

# 

# http://www.gnu.org/licenses/ 

#***************************************************************************** 

from __future__ import print_function, absolute_import 

import math 

from . import shapes 

from .base import PrimitiveObject, point_list_bounding_box 

from sage.rings.real_double import RDF 

from sage.modules.free_module_element import vector 

from sage.misc.decorators import options, rename_keyword 

from sage.arith.srange import srange 

from .texture import Texture 

TACHYON_PIXEL = 1/200.0 

from .shapes import Text, Sphere 

from sage.structure.element import is_Vector 

def line3d(points, thickness=1, radius=None, arrow_head=False, **kwds): 

r""" 

Draw a 3d line joining a sequence of points. 

One may specify either a thickness or radius. If a thickness is 

specified, this line will have a constant diameter regardless of 

scaling and zooming. If a radius is specified, it will behave as a 

series of cylinders. 

INPUT: 

- ``points`` -- a list of at least 2 points 

- ``thickness`` -- (default: 1) 

- ``radius`` -- (default: None) 

- ``arrow_head`` -- (default: False) 

- ``color`` -- a string (``"red"``, ``"green"`` etc) 

or a tuple (r, g, b) with r, g, b numbers between 0 and 1 

- ``opacity`` -- (default: 1) if less than 1 then is 

transparent 

EXAMPLES: 

A line in 3-space:: 

sage: line3d([(1,2,3), (1,0,-2), (3,1,4), (2,1,-2)]) 

Graphics3d Object 

The same line but red:: 

sage: line3d([(1,2,3), (1,0,-2), (3,1,4), (2,1,-2)], color='red') 

Graphics3d Object 

The points of the line provided as a numpy array:: 

sage: import numpy 

sage: line3d(numpy.array([(1,2,3), (1,0,-2), (3,1,4), (2,1,-2)])) 

Graphics3d Object 

A transparent thick green line and a little blue line:: 

sage: line3d([(0,0,0), (1,1,1), (1,0,2)], opacity=0.5, radius=0.1, 

....: color='green') + line3d([(0,1,0), (1,0,2)]) 

Graphics3d Object 

A Dodecahedral complex of 5 tetrahedra (a more elaborate example 

from Peter Jipsen):: 

sage: def tetra(col): 

....: return line3d([(0,0,1), (2*sqrt(2.)/3,0,-1./3), (-sqrt(2.)/3, sqrt(6.)/3,-1./3),\ 

....: (-sqrt(2.)/3,-sqrt(6.)/3,-1./3), (0,0,1), (-sqrt(2.)/3, sqrt(6.)/3,-1./3),\ 

....: (-sqrt(2.)/3,-sqrt(6.)/3,-1./3), (2*sqrt(2.)/3,0,-1./3)],\ 

....: color=col, thickness=10, aspect_ratio=[1,1,1]) 

sage: v = (sqrt(5.)/2-5/6, 5/6*sqrt(3.)-sqrt(15.)/2, sqrt(5.)/3) 

sage: t = acos(sqrt(5.)/3)/2 

sage: t1 = tetra('blue').rotateZ(t) 

sage: t2 = tetra('red').rotateZ(t).rotate(v,2*pi/5) 

sage: t3 = tetra('green').rotateZ(t).rotate(v,4*pi/5) 

sage: t4 = tetra('yellow').rotateZ(t).rotate(v,6*pi/5) 

sage: t5 = tetra('orange').rotateZ(t).rotate(v,8*pi/5) 

sage: show(t1+t2+t3+t4+t5, frame=False) 

TESTS: 

Copies are made of the input list, so the input list does not change:: 

sage: mypoints = [vector([1,2,3]), vector([4,5,6])] 

sage: type(mypoints[0]) 

<... 'sage.modules.vector_integer_dense.Vector_integer_dense'> 

sage: L = line3d(mypoints) 

sage: type(mypoints[0]) 

<... 'sage.modules.vector_integer_dense.Vector_integer_dense'> 

The copies are converted to a list, so we can pass in immutable objects too:: 

sage: L = line3d(((0,0,0),(1,2,3))) 

This function should work for anything than can be turned into a 

list, such as iterators and such (see :trac:`10478`):: 

sage: line3d(iter([(0,0,0), (sqrt(3), 2, 4)])) 

Graphics3d Object 

sage: line3d((x, x^2, x^3) for x in range(5)) 

Graphics3d Object 

sage: from builtins import zip 

sage: line3d(zip([2,3,5,7], [11, 13, 17, 19], [-1, -2, -3, -4])) 

Graphics3d Object 

""" 

points = list(points) 

if len(points) < 2: 

raise ValueError("there must be at least 2 points") 

for i in range(len(points)): 

x, y, z = points[i] 

points[i] = float(x), float(y), float(z) 

if radius is None: 

L = Line(points, thickness=thickness, arrow_head=arrow_head, **kwds) 

L._set_extra_kwds(kwds) 

L._extra_kwds['thickness'] = thickness # remove this line if json_repr is defined 

return L 

else: 

v = [] 

if 'texture' in kwds: 

kwds = kwds.copy() 

texture = kwds.pop('texture') 

else: 

texture = Texture(kwds) 

for i in range(len(points) - 1): 

line = shapes.arrow3d if i == len(points)-2 and arrow_head else shapes.LineSegment 

v.append(line(points[i], points[i+1], texture=texture, radius=radius, **kwds)) 

w = sum(v) 

w._set_extra_kwds(kwds) 

return w 

@options(opacity=1, color="blue", aspect_ratio=[1,1,1], thickness=2) 

def bezier3d(path, **options): 

""" 

Draw a 3-dimensional bezier path. 

Input is similar to bezier_path, but each point in the path and 

each control point is required to have 3 coordinates. 

INPUT: 

- ``path`` -- a list of curves, which each is a list of points. See further 

detail below. 

- ``thickness`` -- (default: 2) 

- ``color`` -- a string (``"red"``, ``"green"`` etc) 

or a tuple (r, g, b) with r, g, b numbers between 0 and 1 

- ``opacity`` -- (default: 1) if less than 1 then is 

transparent 

- ``aspect_ratio`` -- (default:[1,1,1]) 

The path is a list of curves, and each curve is a list of points. 

Each point is a tuple (x,y,z). 

The first curve contains the endpoints as the first and last point 

in the list. All other curves assume a starting point given by the 

last entry in the preceding list, and take the last point in the list 

as their opposite endpoint. A curve can have 0, 1 or 2 control points 

listed between the endpoints. In the input example for path below, 

the first and second curves have 2 control points, the third has one, 

and the fourth has no control points:: 

path = [[p1, c1, c2, p2], [c3, c4, p3], [c5, p4], [p5], ...] 

In the case of no control points, a straight line will be drawn 

between the two endpoints. If one control point is supplied, then 

the curve at each of the endpoints will be tangent to the line from 

that endpoint to the control point. Similarly, in the case of two 

control points, at each endpoint the curve will be tangent to the line 

connecting that endpoint with the control point immediately after or 

immediately preceding it in the list. 

So in our example above, the curve between p1 and p2 is tangent to the 

line through p1 and c1 at p1, and tangent to the line through p2 and c2 

at p2. Similarly, the curve between p2 and p3 is tangent to line(p2,c3) 

at p2 and tangent to line(p3,c4) at p3. Curve(p3,p4) is tangent to 

line(p3,c5) at p3 and tangent to line(p4,c5) at p4. Curve(p4,p5) is a 

straight line. 

EXAMPLES:: 

sage: path = [[(0,0,0),(.5,.1,.2),(.75,3,-1),(1,1,0)],[(.5,1,.2),(1,.5,0)],[(.7,.2,.5)]] 

sage: b = bezier3d(path, color='green') 

sage: b 

Graphics3d Object 

To construct a simple curve, create a list containing a single list:: 

sage: path = [[(0,0,0),(1,0,0),(0,1,0),(0,1,1)]] 

sage: curve = bezier3d(path, thickness=5, color='blue') 

sage: curve 

Graphics3d Object 

""" 

from . import parametric_plot3d as P3D 

from sage.modules.free_module_element import vector 

from sage.symbolic.ring import SR 

p0 = vector(path[0][-1]) 

t = SR.var('t') 

if len(path[0]) > 2: 

B = (1-t)**3*vector(path[0][0])+3*t*(1-t)**2*vector(path[0][1])+3*t**2*(1-t)*vector(path[0][-2])+t**3*p0 

G = P3D.parametric_plot3d(list(B), (0, 1), color=options['color'], aspect_ratio=options['aspect_ratio'], thickness=options['thickness'], opacity=options['opacity']) 

else: 

G = line3d([path[0][0], p0], color=options['color'], thickness=options['thickness'], opacity=options['opacity']) 

for curve in path[1:]: 

if len(curve) > 1: 

p1 = vector(curve[0]) 

p2 = vector(curve[-2]) 

p3 = vector(curve[-1]) 

B = (1-t)**3*p0+3*t*(1-t)**2*p1+3*t**2*(1-t)*p2+t**3*p3 

G += P3D.parametric_plot3d(list(B), (0, 1), color=options['color'], aspect_ratio=options['aspect_ratio'], thickness=options['thickness'], opacity=options['opacity']) 

else: 

G += line3d([p0,curve[0]], color=options['color'], thickness=options['thickness'], opacity=options['opacity']) 

p0 = curve[-1] 

return G 

@rename_keyword(alpha='opacity') 

@options(opacity=1, color=(0,0,1)) 

def polygon3d(points, **options): 

""" 

Draw a polygon in 3d. 

INPUT: 

- ``points`` -- the vertices of the polygon 

Type ``polygon3d.options`` for a dictionary of the default 

options for polygons. You can change this to change 

the defaults for all future polygons. Use ``polygon3d.reset()`` 

to reset to the default options. 

EXAMPLES: 

A simple triangle:: 

sage: polygon3d([[0,0,0], [1,2,3], [3,0,0]]) 

Graphics3d Object 

Some modern art -- a random polygon:: 

sage: v = [(randrange(-5,5), randrange(-5,5), randrange(-5, 5)) for _ in range(10)] 

sage: polygon3d(v) 

Graphics3d Object 

A bent transparent green triangle:: 

sage: polygon3d([[1, 2, 3], [0,1,0], [1,0,1], [3,0,0]], color=(0,1,0), alpha=0.7) 

Graphics3d Object 

""" 

from sage.plot.plot3d.index_face_set import IndexFaceSet 

return IndexFaceSet([range(len(points))], points, **options) 

@options(opacity=1, color=(0,0,1)) 

def polygons3d(faces, points, **options): 

""" 

Draw the union of several polygons in 3d. 

Useful to plot a polyhedron as just one ``IndexFaceSet``. 

INPUT: 

- ``faces`` -- list of faces, every face given by the list 

of indices of its vertices 

- ``points`` -- coordinates of the vertices in the union 

EXAMPLES: 

Two adjacent triangles:: 

sage: f = [[0,1,2],[1,2,3]] 

sage: v = [(-1,0,0),(0,1,1),(0,-1,1),(1,0,0)] 

sage: polygons3d(f, v, color='red') 

Graphics3d Object 

""" 

from sage.plot.plot3d.index_face_set import IndexFaceSet 

return IndexFaceSet(faces, points, **options) 

def frame3d(lower_left, upper_right, **kwds): 

""" 

Draw a frame in 3-D. 

Primarily used as a helper function for creating frames for 3-D 

graphics viewing. 

INPUT: 

- ``lower_left`` -- the lower left corner of the frame, as a 

list, tuple, or vector. 

- ``upper_right`` -- the upper right corner of the frame, as a 

list, tuple, or vector. 

Type ``line3d.options`` for a dictionary of the default 

options for lines, which are also available. 

EXAMPLES: 

A frame:: 

sage: from sage.plot.plot3d.shapes2 import frame3d 

sage: frame3d([1,3,2],vector([2,5,4]),color='red') 

Graphics3d Object 

This is usually used for making an actual plot:: 

sage: y = var('y') 

sage: plot3d(sin(x^2+y^2),(x,0,pi),(y,0,pi)) 

Graphics3d Object 

""" 

x0,y0,z0 = lower_left 

x1,y1,z1 = upper_right 

L1 = line3d([(x0,y0,z0), (x0,y1,z0), (x1,y1,z0), (x1,y0,z0), (x0,y0,z0), # top square 

(x0,y0,z1), (x0,y1,z1), (x1,y1,z1), (x1,y0,z1), (x0,y0,z1)], # bottom square 

**kwds) 

# 3 additional lines joining top to bottom 

v2 = line3d([(x0,y1,z0), (x0,y1,z1)], **kwds) 

v3 = line3d([(x1,y0,z0), (x1,y0,z1)], **kwds) 

v4 = line3d([(x1,y1,z0), (x1,y1,z1)], **kwds) 

F = L1 + v2 + v3 + v4 

F._set_extra_kwds(kwds) 

return F 

def frame_labels(lower_left, upper_right, 

label_lower_left, label_upper_right, eps = 1, 

**kwds): 

""" 

Draw correct labels for a given frame in 3-D. 

Primarily used as a helper function for creating frames for 3-D 

graphics viewing - do not use directly unless you know what you 

are doing! 

INPUT: 

- ``lower_left`` -- the lower left corner of the frame, as a 

list, tuple, or vector. 

- ``upper_right`` -- the upper right corner of the frame, as a 

list, tuple, or vector. 

- ``label_lower_left`` -- the label for the lower left corner 

of the frame, as a list, tuple, or vector. This label must actually 

have all coordinates less than the coordinates of the other label. 

- ``label_upper_right`` -- the label for the upper right corner 

of the frame, as a list, tuple, or vector. This label must actually 

have all coordinates greater than the coordinates of the other label. 

- ``eps`` -- (default: 1) a parameter for how far away from the frame 

to put the labels. 

Type ``line3d.options`` for a dictionary of the default 

options for lines, which are also available. 

EXAMPLES: 

We can use it directly:: 

sage: from sage.plot.plot3d.shapes2 import frame_labels 

sage: frame_labels([1,2,3],[4,5,6],[1,2,3],[4,5,6]) 

Graphics3d Object 

This is usually used for making an actual plot:: 

sage: y = var('y') 

sage: P = plot3d(sin(x^2+y^2),(x,0,pi),(y,0,pi)) 

sage: a,b = P._rescale_for_frame_aspect_ratio_and_zoom(1.0,[1,1,1],1) 

sage: F = frame_labels(a,b,*P._box_for_aspect_ratio("automatic",a,b)) 

sage: F.jmol_repr(F.default_render_params())[0] 

[['select atomno = 1', 'color atom [76,76,76]', 'label "0.0"']] 

TESTS:: 

sage: frame_labels([1,2,3],[4,5,6],[1,2,3],[1,3,4]) 

Traceback (most recent call last): 

... 

ValueError: Ensure the upper right labels are above and to the right of the lower left labels. 

""" 

x0,y0,z0 = lower_left 

x1,y1,z1 = upper_right 

lx0,ly0,lz0 = label_lower_left 

lx1,ly1,lz1 = label_upper_right 

if (lx1 - lx0) <= 0 or (ly1 - ly0) <= 0 or (lz1 - lz0) <= 0: 

raise ValueError("Ensure the upper right labels are above and to the right of the lower left labels.") 

# Helper function for formatting the frame labels 

from math import log 

log10 = log(10) 

nd = lambda a: int(log(a)/log10) 

def fmt_string(a): 

b = a/2.0 

if b >= 1: 

return "%.1f" 

n = max(0, 2 - nd(a/2.0)) 

return "%%.%sf"%n 

# Slightly faster than mean for this situation 

def avg(a,b): 

return (a+b)/2.0 

color = (0.3,0.3,0.3) 

fmt = fmt_string(lx1 - lx0) 

T = Text(fmt%lx0, color=color).translate((x0,y0-eps,z0)) 

T += Text(fmt%avg(lx0,lx1), color=color).translate((avg(x0,x1),y0-eps,z0)) 

T += Text(fmt%lx1, color=color).translate((x1,y0-eps,z0)) 

fmt = fmt_string(ly1 - ly0) 

T += Text(fmt%ly0, color=color).translate((x1+eps,y0,z0)) 

T += Text(fmt%avg(ly0,ly1), color=color).translate((x1+eps,avg(y0,y1),z0)) 

T += Text(fmt%ly1, color=color).translate((x1+eps,y1,z0)) 

fmt = fmt_string(lz1 - lz0) 

T += Text(fmt%lz0, color=color).translate((x0-eps,y0,z0)) 

T += Text(fmt%avg(lz0,lz1), color=color).translate((x0-eps,y0,avg(z0,z1))) 

T += Text(fmt%lz1, color=color).translate((x0-eps,y0,z1)) 

return T 

def ruler(start, end, ticks=4, sub_ticks=4, absolute=False, snap=False, **kwds): 

""" 

Draw a ruler in 3-D, with major and minor ticks. 

INPUT: 

- ``start`` -- the beginning of the ruler, as a list, 

tuple, or vector. 

- ``end`` -- the end of the ruler, as a list, tuple, 

or vector. 

- ``ticks`` -- (default: 4) the number of major ticks 

shown on the ruler. 

- ``sub_ticks`` -- (default: 4) the number of shown 

subdivisions between each major tick. 

- ``absolute`` -- (default: ``False``) if ``True``, makes a huge ruler 

in the direction of an axis. 

- ``snap`` -- (default: ``False``) if ``True``, snaps to an implied 

grid. 

Type ``line3d.options`` for a dictionary of the default 

options for lines, which are also available. 

EXAMPLES: 

A ruler:: 

sage: from sage.plot.plot3d.shapes2 import ruler 

sage: R = ruler([1,2,3],vector([2,3,4])); R 

Graphics3d Object 

A ruler with some options:: 

sage: R = ruler([1,2,3],vector([2,3,4]),ticks=6, sub_ticks=2, color='red'); R 

Graphics3d Object 

The keyword ``snap`` makes the ticks not necessarily coincide 

with the ruler:: 

sage: ruler([1,2,3],vector([1,2,4]),snap=True) 

Graphics3d Object 

The keyword ``absolute`` makes a huge ruler in one of the axis 

directions:: 

sage: ruler([1,2,3],vector([1,2,4]),absolute=True) 

Graphics3d Object 

TESTS:: 

sage: ruler([1,2,3],vector([1,3,4]),absolute=True) 

Traceback (most recent call last): 

... 

ValueError: Absolute rulers only valid for axis-aligned paths 

""" 

start = vector(RDF, start) 

end = vector(RDF, end) 

dir = end - start 

dist = math.sqrt(dir.dot_product(dir)) 

dir /= dist 

one_tick = dist/ticks * 1.414 

unit = 10 ** math.floor(math.log(dist/ticks, 10)) 

if unit * 5 < one_tick: 

unit *= 5 

elif unit * 2 < one_tick: 

unit *= 2 

if dir[0]: 

tick = dir.cross_product(vector(RDF, (0,0,-dist/30))) 

elif dir[1]: 

tick = dir.cross_product(vector(RDF, (0,0,dist/30))) 

else: 

tick = vector(RDF, (dist/30,0,0)) 

if snap: 

for i in range(3): 

start[i] = unit * math.floor(start[i]/unit + 1e-5) 

end[i] = unit * math.ceil(end[i]/unit - 1e-5) 

if absolute: 

if dir[0]*dir[1] or dir[1]*dir[2] or dir[0]*dir[2]: 

raise ValueError("Absolute rulers only valid for axis-aligned paths") 

m = max(dir[0], dir[1], dir[2]) 

if dir[0] == m: 

off = start[0] 

elif dir[1] == m: 

off = start[1] 

else: 

off = start[2] 

first_tick = unit * math.ceil(off/unit - 1e-5) - off 

else: 

off = 0 

first_tick = 0 

ruler = shapes.LineSegment(start, end, **kwds) 

for k in range(1, int(sub_ticks * first_tick/unit)): 

P = start + dir*(k*unit/sub_ticks) 

ruler += shapes.LineSegment(P, P + tick/2, **kwds) 

for d in srange(first_tick, dist + unit/(sub_ticks+1), unit): 

P = start + dir*d 

ruler += shapes.LineSegment(P, P + tick, **kwds) 

ruler += shapes.Text(str(d+off), **kwds).translate(P - tick) 

if dist - d < unit: 

sub_ticks = int(sub_ticks * (dist - d)/unit) 

for k in range(1, sub_ticks): 

P += dir * (unit/sub_ticks) 

ruler += shapes.LineSegment(P, P + tick/2, **kwds) 

return ruler 

def ruler_frame(lower_left, upper_right, ticks=4, sub_ticks=4, **kwds): 

""" 

Draw a frame made of 3-D rulers, with major and minor ticks. 

INPUT: 

- ``lower_left`` -- the lower left corner of the frame, as a 

list, tuple, or vector. 

- ``upper_right`` -- the upper right corner of the frame, as a 

list, tuple, or vector. 

- ``ticks`` -- (default: 4) the number of major ticks 

shown on each ruler. 

- ``sub_ticks`` -- (default: 4) the number of shown 

subdivisions between each major tick. 

Type ``line3d.options`` for a dictionary of the default 

options for lines, which are also available. 

EXAMPLES: 

A ruler frame:: 

sage: from sage.plot.plot3d.shapes2 import ruler_frame 

sage: F = ruler_frame([1,2,3],vector([2,3,4])); F 

Graphics3d Object 

A ruler frame with some options:: 

sage: F = ruler_frame([1,2,3],vector([2,3,4]),ticks=6, sub_ticks=2, color='red'); F 

Graphics3d Object 

""" 

return ruler(lower_left, (upper_right[0], lower_left[1], lower_left[2]), ticks=ticks, sub_ticks=sub_ticks, absolute=True, **kwds) \ 

+ ruler(lower_left, (lower_left[0], upper_right[1], lower_left[2]), ticks=ticks, sub_ticks=sub_ticks, absolute=True, **kwds) \ 

+ ruler(lower_left, (lower_left[0], lower_left[1], upper_right[2]), ticks=ticks, sub_ticks=sub_ticks, absolute=True, **kwds) 

########################### 

def sphere(center=(0,0,0), size=1, **kwds): 

r""" 

Return a plot of a sphere of radius ``size`` centered at 

`(x,y,z)`. 

INPUT: 

- `(x,y,z)` -- center (default: (0,0,0)) 

- ``size`` -- the radius (default: 1) 

EXAMPLES: A simple sphere:: 

sage: sphere() 

Graphics3d Object 

Two spheres touching:: 

sage: sphere(center=(-1,0,0)) + sphere(center=(1,0,0), aspect_ratio=[1,1,1]) 

Graphics3d Object 

Spheres of radii 1 and 2 one stuck into the other:: 

sage: sphere(color='orange') + sphere(color=(0,0,0.3), 

....: center=(0,0,-2),size=2,opacity=0.9) 

Graphics3d Object 

We draw a transparent sphere on a saddle. :: 

sage: u,v = var('u v') 

sage: saddle = plot3d(u^2 - v^2, (u,-2,2), (v,-2,2)) 

sage: sphere((0,0,1), color='red', opacity=0.5, aspect_ratio=[1,1,1]) + saddle 

Graphics3d Object 

TESTS:: 

sage: T = sage.plot.plot3d.texture.Texture('red') 

sage: S = sphere(texture=T) 

sage: T in S.texture_set() 

True 

""" 

kwds['texture'] = Texture(**kwds) 

G = Sphere(size, **kwds) 

H = G.translate(center) 

H._set_extra_kwds(kwds) 

return H 

def text3d(txt, x_y_z, **kwds): 

r""" 

Display 3d text. 

INPUT: 

- ``txt`` -- some text 

- ``(x,y,z)`` -- position tuple `(x,y,z)` 

- ``**kwds`` -- standard 3d graphics options 

.. note:: 

There is no way to change the font size or opacity yet. 

EXAMPLES: 

We write the word Sage in red at position (1,2,3):: 

sage: text3d("Sage", (1,2,3), color=(0.5,0,0)) 

Graphics3d Object 

We draw a multicolor spiral of numbers:: 

sage: sum([text3d('%.1f'%n, (cos(n),sin(n),n), color=(n/2,1-n/2,0)) 

....: for n in [0,0.2,..,8]]) 

Graphics3d Object 

Another example:: 

sage: text3d("Sage is really neat!!",(2,12,1)) 

Graphics3d Object 

And in 3d in two places:: 

sage: text3d("Sage is...",(2,12,1), color=(1,0,0)) + text3d("quite powerful!!",(4,10,0), color=(0,0,1)) 

Graphics3d Object 

""" 

(x, y, z) = x_y_z 

if 'color' not in kwds and 'rgbcolor' not in kwds: 

kwds['color'] = (0, 0, 0) 

G = Text(txt, **kwds).translate((x, y, z)) 

G._set_extra_kwds(kwds) 

return G 

class Point(PrimitiveObject): 

""" 

Create a position in 3-space, represented by a sphere of fixed 

size. 

INPUT: 

- ``center`` -- point (3-tuple) 

- ``size`` -- (default: 1) 

EXAMPLES: 

We normally access this via the ``point3d`` function. Note that extra 

keywords are correctly used:: 

sage: point3d((4,3,2),size=2,color='red',opacity=.5) 

Graphics3d Object 

""" 

def __init__(self, center, size=1, **kwds): 

""" 

Create the graphics primitive :class:`Point` in 3-D. 

See the docstring of this class for full documentation. 

EXAMPLES:: 

sage: from sage.plot.plot3d.shapes2 import Point 

sage: P = Point((1,2,3),2) 

sage: P.loc 

(1.0, 2.0, 3.0) 

""" 

PrimitiveObject.__init__(self, **kwds) 

self.loc = (float(center[0]), float(center[1]), float(center[2])) 

self.size = size 

self._set_extra_kwds(kwds) 

def bounding_box(self): 

""" 

Returns the lower and upper corners of a 3-D bounding box for ``self``. 

This is used for rendering and ``self`` should fit entirely within this 

box. In this case, we simply return the center of the point. 

TESTS:: 

sage: P = point3d((-3,2,10),size=7) 

sage: P.bounding_box() 

((-3.0, 2.0, 10.0), (-3.0, 2.0, 10.0)) 

""" 

return self.loc, self.loc 

def tachyon_repr(self, render_params): 

""" 

Return representation of the point suitable for plotting 

using the Tachyon ray tracer. 

TESTS:: 

sage: P = point3d((1,2,3),size=3,color='purple') 

sage: P.tachyon_repr(P.default_render_params()) 

'Sphere center 1.0 2.0 3.0 Rad 0.015 texture...' 

""" 

transform = render_params.transform 

if transform is None: 

cen = self.loc 

else: 

cen = transform.transform_point(self.loc) 

return "Sphere center %s %s %s Rad %s %s" % (cen[0], cen[1], cen[2], self.size * TACHYON_PIXEL, self.texture.id) 

def obj_repr(self, render_params): 

""" 

Return complete representation of the point as a sphere. 

TESTS:: 

sage: P = point3d((1,2,3),size=3,color='purple') 

sage: P.obj_repr(P.default_render_params())[0][0:2] 

['g obj_1', 'usemtl texture...'] 

""" 

T = render_params.transform 

if T is None: 

from . import transform 

T = transform.Transformation() 

render_params.push_transform(~T) 

S = shapes.Sphere(self.size / 200.0).translate(T(self.loc)) 

cmds = S.obj_repr(render_params) 

render_params.pop_transform() 

return cmds 

def jmol_repr(self, render_params): 

r""" 

Return representation of the object suitable for plotting 

using Jmol. 

TESTS:: 

sage: P = point3d((1,2,3),size=3,color='purple') 

sage: P.jmol_repr(P.default_render_params()) 

['draw point_1 DIAMETER 3 {1.0 2.0 3.0}\ncolor $point_1 [128,0,128]'] 

""" 

name = render_params.unique_name('point') 

transform = render_params.transform 

cen = self.loc if transform is None else transform(self.loc) 

return ["draw %s DIAMETER %s {%s %s %s}\n%s" % (name, int(self.size), cen[0], cen[1], cen[2], self.texture.jmol_str('$' + name))] 

class Line(PrimitiveObject): 

r""" 

Draw a 3d line joining a sequence of points. 

This line has a fixed diameter unaffected by transformations and 

zooming. It may be smoothed if ``corner_cutoff < 1``. 

INPUT: 

- ``points`` -- list of points to pass through 

- ``thickness`` -- (optional, default 5) diameter of the line 

- ``corner_cutoff`` -- (optional, default 0.5) threshold for 

smoothing (see :meth:`corners`). 

- ``arrow_head`` -- (optional, default ``False``) if ``True`` make 

this curve into an arrow 

The parameter ``corner_cutoff`` is a bound for the cosine of the 

angle made by two successive segments. This angle is close to `0` 

(and the cosine close to 1) if the two successive segments are 

almost aligned and close to `\pi` (and the cosine close to -1) if 

the path has a strong peak. If the cosine is smaller than the 

bound (which means a sharper peak) then no smoothing is done. 

EXAMPLES:: 

sage: from sage.plot.plot3d.shapes2 import Line 

sage: Line([(i*math.sin(i), i*math.cos(i), i/3) for i in range(30)], arrow_head=True) 

Graphics3d Object 

Smooth angles less than 90 degrees:: 

sage: Line([(0,0,0),(1,0,0),(2,1,0),(0,1,0)], corner_cutoff=0) 

Graphics3d Object 

Make sure that the ``corner_cutoff`` keyword works (:trac:`3859`):: 

sage: N = 11 

sage: c = 0.4 

sage: sum([Line([(i,1,0), (i,0,0), (i,cos(2*pi*i/N), sin(2*pi*i/N))], 

....: corner_cutoff=c, 

....: color='red' if -cos(2*pi*i/N)<=c else 'blue') 

....: for i in range(N+1)]) 

Graphics3d Object 

""" 

def __init__(self, points, thickness=5, corner_cutoff=0.5, 

arrow_head=False, **kwds): 

""" 

Create the graphics primitive :class:`Line` in 3-D. 

See the docstring of this class for full documentation. 

EXAMPLES:: 

sage: from sage.plot.plot3d.shapes2 import Line 

sage: P = Line([(1,2,3),(1,2,2),(-1,2,2),(-1,3,2)],thickness=6,corner_cutoff=.2) 

sage: P.points, P.arrow_head 

([(1, 2, 3), (1, 2, 2), (-1, 2, 2), (-1, 3, 2)], False) 

""" 

if len(points) < 2: 

raise ValueError("there must be at least 2 points") 

PrimitiveObject.__init__(self, **kwds) 

self.points = points 

self.thickness = thickness 

self.corner_cutoff = corner_cutoff 

self.arrow_head = arrow_head 

def bounding_box(self): 

""" 

Return the lower and upper corners of a 3-D bounding box for ``self``. 

This is used for rendering and ``self`` should fit entirely within this 

box. In this case, we return the highest and lowest values of each 

coordinate among all points. 

TESTS:: 

sage: from sage.plot.plot3d.shapes2 import Line 

sage: L = Line([(i,i^2-1,-2*ln(i)) for i in [10,20,30]]) 

sage: L.bounding_box() 

((10.0, 99.0, -6.802394763324311), 

(30.0, 899.0, -4.605170185988092)) 

""" 

try: 

return self.__bounding_box 

except AttributeError: 

self.__bounding_box = point_list_bounding_box(self.points) 

return self.__bounding_box 

def tachyon_repr(self, render_params): 

""" 

Return representation of the line suitable for plotting 

using the Tachyon ray tracer. 

TESTS:: 

sage: L = line3d([(cos(i),sin(i),i^2) for i in srange(0,10,.01)],color='red') 

sage: L.tachyon_repr(L.default_render_params())[0] 

'FCylinder base 1.0 0.0 0.0 apex 0.999950000417 0.00999983333417 0.0001 rad 0.005 texture...' 

""" 

T = render_params.transform 

cmds = [] 

px, py, pz = self.points[0] if T is None else T(self.points[0]) 

radius = self.thickness * TACHYON_PIXEL 

for P in self.points[1:]: 

x, y, z = P if T is None else T(P) 

if self.arrow_head and P is self.points[-1]: 

A = shapes.arrow3d((px, py, pz), (x, y, z), radius = radius, texture = self.texture) 

render_params.push_transform(~T) 

cmds.append(A.tachyon_repr(render_params)) 

render_params.pop_transform() 

else: 

cmds.append("FCylinder base %s %s %s apex %s %s %s rad %s %s" % (px, py, pz, 

x, y, z, 

radius, 

self.texture.id)) 

px, py, pz = x, y, z 

return cmds 

def obj_repr(self, render_params): 

""" 

Return complete representation of the line as an object. 

TESTS:: 

sage: from sage.plot.plot3d.shapes2 import Line 

sage: L = Line([(cos(i),sin(i),i^2) for i in srange(0,10,.01)],color='red') 

sage: L.obj_repr(L.default_render_params())[0][0][0][2][:3] 

['v 0.99995 0.00999983 0.0001', 

'v 1.02376 0.010195 -0.00750607', 

'v 1.00007 0.0102504 -0.0248984'] 

""" 

T = render_params.transform 

if T is None: 

from . import transform 

T = transform.Transformation() 

render_params.push_transform(~T) 

L = line3d([T(P) for P in self.points], radius=self.thickness / 200.0, arrow_head=self.arrow_head, texture=self.texture) 

cmds = L.obj_repr(render_params) 

render_params.pop_transform() 

return cmds 

def jmol_repr(self, render_params): 

r""" 

Return representation of the object suitable for plotting 

using Jmol. 

TESTS:: 

sage: L = line3d([(cos(i),sin(i),i^2) for i in srange(0,10,.01)],color='red') 

sage: L.jmol_repr(L.default_render_params())[0][:42] 

'draw line_1 diameter 1 curve {1.0 0.0 0.0}' 

""" 

T = render_params.transform 

corners = self.corners(max_len=255) # hardcoded limit in jmol 

last_corner = corners[-1] 

corners = set(corners) 

cmds = [] 

cmd = None 

for P in self.points: 

TP = P if T is None else T(P) 

if P in corners: 

if cmd: 

cmds.append(cmd + " {%s %s %s} " % TP) 

cmds.append(self.texture.jmol_str('$'+name)) 

type = 'arrow' if self.arrow_head and P is last_corner else 'curve' 

name = render_params.unique_name('line') 

cmd = "draw %s diameter %s %s {%s %s %s} " % (name, int(self.thickness), type, TP[0], TP[1], TP[2]) 

else: 

cmd += " {%s %s %s} " % TP 

cmds.append(cmd) 

cmds.append(self.texture.jmol_str('$'+name)) 

return cmds