# Graphics Capabilities of Sage

Many of the example of this section are taken from Sage Documentation : http://sagemath.org/doc

## 2d plots

The following plotting functions are supported:

• plot() - plot of a function or other Sage object (e.g., elliptic curve).
• parametric_plot()
• implicit_plot()
• polar_plot()
• region_plot()
• list_plot()
• scatter_plot()
• bar_chart()
• contour_plot()
• density_plot()
• plot_vector_field()
• plot_slope_field()
• matrix_plot()
• complex_plot()
• graphics_array()

Dessiner une fonction $\mathbb{R}\mapsto \mathbb{R}$ : la commande plot:

sage: x,y = var('x y')
sage: plot_vector_field((sin(x), cos(y)), (x,-3,3), (y,-3,3))

sage: f(x) = x^4 - 1
sage: complex_plot(f, (-2,2), (-2,2))


## 2d primitives

The following graphics primitives are supported:

• arrow() - an arrow from a min point to a max point.
• circle() - a circle with given radius
• ellipse() - an ellipse with given rad and angle
• arc() - an arc of a circle or an ellipse
• disk() - a filled disk (i.e. a sector or wedge of a circle)
• line() - a line determined by a sequence of points (this need not be straight!)
• point() - a point
• text() - some text
• polygon() - a filled polygon
sage: f = sin(1/x)
sage: P = plot(f, -10, 10, color='red')
sage: P

sage: Q = line([(3,0.9), (7,0.9), (7,1.1), (3,1.1), (3,0.9)], color='green')
sage: Q

sage: R = text('$f(x) = \\sin(\\frac{1}{x})$', (5,1))
sage: R

sage: P + Q + R


For more examples on 2D ploting, visit: http://sagemath.org/doc/reference/plotting.html

## 3d plots

Dessiner une fonction $\mathbb{R}^2\mapsto \mathbb{R}$ : la commande plot3d:

sage: def f(x, y): return x^2 + y^2
sage: plot3d(f, (-10,10), (-10,10), viewer='tachyon')


the same with Jmol:

sage: plot3d(f, (-10,10), (-10,10))


A hyperboloid:

sage: x, y, z = var('x, y, z')
sage: implicit_plot3d(x^2 + y^2 - z^2 -0.3, (x, -2, 2), (y, -2, 2), (z, -1.8, 1.8))


A torus:

sage: implicit_plot3d((sqrt(x*x+y*y)-3)^2 + z*z - 1, (x, -4, 4), (y, -4, 4), (z, -1, 1))


## 3D primitives

Two spheres touching:

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


A transparent thick green line and a little blue line:

sage: A = line3d([(0,0,0), (1,1,1), (1,0,2)], opacity=0.5, radius=0.1, color='green')
sage: B = line3d([(0,1,0), (1,0,2)])
sage: A + B

sage: from sage.plot.plot3d.shapes2 import ruler
sage: R = ruler([1,2,3],vector([2,3,4]),ticks=6, sub_ticks=2, color='red'); R


For more examples on 3D ploting, visit: http://sagemath.org/doc/reference/plot3d.html

# Discrete Geometry using Sage

(The following is not in Sage currently but could be soon if there is an interest).

sage: load /Users/slabbe/Documents/Projets/laber/discrete_object.sage


## Discrete planes

sage: P = Plan([1,3,7])
sage: P
Plan discret de vecteur normal v=(1, 3, 7), d'épaisseur omega=11 et d'intercept h=0.
Bounding Box: ((0, 0, 0), (11, 11, 11))

sage: P.list()
[(0, 0, 1), (0, 1, 0), (0, 1, 1), (0, 2, 0), (0, 3, 0), (1, 0, 0), (1, 0, 1), (1, 1, 0), (1, 1, 1), (1, 2, 0), (1, 3, 0), (2, 0, 0), (2, 0, 1), (2, 1, 0), (2, 2, 0), (2, 3, 0), (3, 0, 0), (3, 0, 1), (3, 1, 0), (3, 2, 0), (4, 0, 0), (4, 0, 1), (4, 1, 0), (4, 2, 0), (5, 0, 0), (5, 1, 0), (5, 2, 0), (6, 0, 0), (6, 1, 0), (7, 0, 0), (7, 1, 0), (8, 0, 0), (8, 1, 0), (9, 0, 0), (10, 0, 0)]

sage: P.show3d()

sage: P = Plan([1,3,5], omega=7, h=20)
sage: P.show3d()


## Discrete 3d lines

Discrete 3d lines defined by cylinder offset:

sage: d = DroiteDiscrete3d_cylinder_offset((1,2,4), offset=1)
sage: d.bounding_box(((0,0,0),(6,6,6)))
sage: d
Droite 3d defined by cylinder offset=1 and direction v=(1, 2, 4).
Bounding Box: ((0, 0, 0), (6, 6, 6))

sage: d.show3d()

sage: d.tikz()


Reveillès Discrete 3d lines:

sage: d = DroiteDiscrete3d_reveilles((12,13,19))
sage: d
Droite 3d de Reveilles v=(12, 13, 19).
Bounding Box: [(0, 0, 0), (20, 20, 20)]

sage: (0,0,0) in d
True

sage: d.show3d()


## Comparing discrete planes

Let's now compare the Reveillès and Toutant 3D discrete lines:

sage: d = DroiteDiscrete3d_reveilles((10,20,29))
sage: t = DroiteDiscrete3d_toutant((10,20,29))
sage: d == t
True


The following example shows that Reveillès and Toutant are different for the vector (1,2,4) and that Toutant points are nearer from the line:

sage: d = DroiteDiscrete3d_reveilles((1,2,4))
sage: t = DroiteDiscrete3d_toutant((1,2,4))
sage: d == t
False
sage: d.list()
[(0, 0, 0), (0, 1, 1), (0, 1, 2), (1, 2, 3), (1, 2, 4)]
sage: t.list()
[(0, 0, 0), (0, 0, 1), (0, 1, 2), (1, 2, 3), (1, 2, 4)]
sage: d.distance((0,1,1))
sqrt(2/7)
sage: d.distance((0,0,1))
sqrt(5/21)


## Intersection of planes

Intersection of two planes:

sage: p = Plan([1,-3,7],h=20)
sage: q = Plan([3,3,-5],h=14)
sage: i = p.intersection(q)
sage: i
Intersection des objets suivants :
Plan discret de vecteur normal v=(1, 3, 7), d'épaisseur omega=11 et d'intercept h=20.
Bounding Box: ((0, 0, 0), (31, 31, 31))
Plan discret de vecteur normal v=(3, 3, 5), d'épaisseur omega=11 et d'intercept h=14.
Bounding Box: ((0, 0, 0), (25, 25, 25))
Bounding Box: ((0, 0, 0), (25, 25, 25))

sage: i.bounding_box(((-25,-25,-25),(25,25,25)))

sage: i.show3d()


# Rauzy fractals

## Construction by projection

(This is will be in version 4.7 of Sage.)

The Rauzy fractal:

sage: s = WordMorphism('1->12,2->13,3->1')
sage: D = s.fixed_point('1')
sage: v = s.pisot_eigenvector_right()
sage: P = WordPaths('123',[(1,0,0),(0,1,0),(0,0,1)])
sage: w = P(D[:200])
sage: w.plot_projection(v) # optional long time (2 s)


The 3d-Rauzy fractal:

sage: s = WordMorphism('1->12,2->13,3->14,4->1')
sage: D = s.fixed_point('1')
sage: v = s.pisot_eigenvector_right()
sage: P = WordPaths('1234',[(1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1)])
sage: w = P(D[:200])
sage: w.plot_projection(v)      # optional long time (1 s)


## Construction by dual morphisms

(This is in Sage since version 4.6.1.)

sage: from sage.combinat.e_one_star import E1Star, Patch, Face
sage: sigma = WordMorphism({1:[1,2], 2:[1,3], 3:[1]})
sage: E = E1Star(sigma)
sage: P = Patch([Face((0,0,0),t) for t in [1,2,3]])
sage: P.plot()

sage: image = E(P, iterations=12)
sage: image.plot()

sage: image.plot3d()