# 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
class sage.plot.plot3d.shapes2.Line(points, thickness=5, corner_cutoff=0.5, arrow_head=False, **kwds)

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 - diameter of the line
• corner_cutoff - threshold for smoothing (see the corners() method) this is the minimum cosine between adjacent segments to smooth
• arrow_head - if True make this curve into an arrow

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)


Smooth angles less than 90 degrees:

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

bounding_box()

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 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))

corners(corner_cutoff=None, max_len=None)

Figures out where the curve turns too sharply to pretend it’s smooth.

INPUT: Maximum cosine of angle between adjacent line segments before adding a corner

OUTPUT: List of points at which to start a new line. This always includes the first point, and never the last.

EXAMPLES:

Every point:

sage: from sage.plot.plot3d.shapes2 import Line
sage: Line([(0,0,0),(1,0,0),(2,1,0),(0,1,0)], corner_cutoff=1).corners()
[(0, 0, 0), (1, 0, 0), (2, 1, 0)]


Greater than 90 degrees:

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


No corners:

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


An intermediate value:

sage: Line([(0,0,0),(1,0,0),(2,1,0),(0,1,0)], corner_cutoff=.5).corners()
[(0, 0, 0), (2, 1, 0)]

jmol_repr(render_params)

Returns 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}'

obj_repr(render_params)

Returns 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 1.00007 0.0102504 -0.0248984', 'v 1.02376 0.010195 -0.00750607']

tachyon_repr(render_params)

Returns 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...'

class sage.plot.plot3d.shapes2.Point(center, size=1, **kwds)

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

INPUT:

• center - point (3-tuple)
• size - (default: 1)

EXAMPLE:

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)

bounding_box()

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))

jmol_repr(render_params)

Returns 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]']

obj_repr(render_params)

Returns 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...']

tachyon_repr(render_params)

Returns 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...'

sage.plot.plot3d.shapes2.bezier3d(path, aspect_ratio=[1, 1, 1], color='blue', opacity=1, thickness=2, **options)

Draws 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 word that describes a color

• 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


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

sage.plot.plot3d.shapes2.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')


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))

sage.plot.plot3d.shapes2.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])


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.


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)
• color - a word that describes a color
• rgbcolor - (r,g,b) with r, g, b between 0 and 1 that describes a color
• 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)])


The same line but red:

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


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)]))


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)])


A Dodecahedral complex of 5 tetrahedrons (a more elaborate examples 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])
<type 'sage.modules.vector_integer_dense.Vector_integer_dense'>
sage: L = line3d(mypoints)
sage: type(mypoints[0])
<type '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 ticket #10478):

sage: line3d(iter([(0,0,0), (sqrt(3), 2, 4)]))
sage: line3d((x, x^2, x^3) for x in range(5))
sage: from itertools import izip; line3d(izip([2,3,5,7], [11, 13, 17, 19], [-1, -2, -3, -4]))

sage.plot.plot3d.shapes2.point3d(v, size=5, **kwds)

Plot a point or list of points in 3d space.

INPUT:

• v – a point or list of points
• size – (default: 5) size of the point (or points)
• color – a word that describes a color
• rgbcolor – (r,g,b) with r, g, b between 0 and 1 that describes a color
• opacity – (default: 1) if less than 1 then is transparent

EXAMPLES:

sage: sum([point3d((i,i^2,i^3), size=5) for i in range(10)])


We check to make sure this works with vectors and other iterables:

sage: pl = point3d([vector(ZZ,(1, 0, 0)), vector(ZZ,(0, 1, 0)), (-1, -1, 0)])
sage: print point(vector((2,3,4)))
Graphics3d Object

sage: c = polytopes.n_cube(3)
sage: v = c.vertices()[0];  v
A vertex at (-1, -1, -1)
sage: print point(v)
Graphics3d Object


We check to make sure the options work:

sage: point3d((4,3,2),size=20,color='red',opacity=.5)


numpy arrays can be provided as input:

sage: import numpy
sage: point3d(numpy.array([1,2,3]))

sage: point3d(numpy.array([[1,2,3], [4,5,6], [7,8,9]]))

sage.plot.plot3d.shapes2.polygon3d(points, color=(0, 0, 1), opacity=1, **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]])


Some modern art – a random polygon:

sage: v = [(randrange(-5,5), randrange(-5,5), randrange(-5, 5)) for _ in range(10)]
sage: polygon3d(v)


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)

sage.plot.plot3d.shapes2.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


A ruler with some options:

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


The keyword snap makes the ticks not necessarily coincide with the ruler:

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


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

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


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

sage.plot.plot3d.shapes2.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


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

sage.plot.plot3d.shapes2.sphere(center=(0, 0, 0), size=1, **kwds)

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()


Two spheres touching:

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


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)


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


TESTS:

sage: T = sage.plot.plot3d.texture.Texture('red')
sage: S = sphere(texture=T)
sage: T in S.texture_set()
True

sage.plot.plot3d.shapes2.text3d(txt, (x, y, z), **kwds)

Display 3d text.

INPUT:

• txt - some text
• (x,y,z) - position
• **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))


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]])


Another example

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


And in 3d in two places:

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


Platonic Solids

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Base classes for 3D Graphics objects and plotting