Plotting functions#

EXAMPLES:

sage: x, y = var('x y')
sage: W = plot3d(sin(pi*((x)^2 + (y)^2))/2, (x, -1, 1), (y, -1, 1),
....:            frame=False, color='purple', opacity=0.8)
sage: S = sphere((0, 0, 0), size=0.3, color='red', aspect_ratio=[1,1,1])
sage: show(W + S, figsize=8)
../../../_images/plot3d-1.svg
sage: def f(x,y):
....:     return math.sin(y^2 + x^2)/math.sqrt(x^2 + y^2 + 0.0001)
sage: P = plot3d(f, (-3, 3),(-3, 3), adaptive=True,
....:            color=rainbow(60, 'rgbtuple'), max_bend=.1, max_depth=15)
sage: P.show()
../../../_images/plot3d-2.svg
sage: def f(x,y):
....:     return math.exp(x/5)*math.sin(y)
....:
sage: P = plot3d(f, (-5, 5), (-5, 5), adaptive=True, color=['red', 'yellow'])
sage: from sage.plot.plot3d.plot3d import axes
sage: S = P + axes(6, color='black')
sage: S.show()
../../../_images/plot3d-3.svg

Here is an example using a colormap and a color function c:

sage: x, y = var('x y')
sage: cm = colormaps.hsv
sage: def c(x, y): return float((x + y + x*y)/15) % 1
sage: plot3d(x*x + y*y, (x, -4, 4), (y, -4, 4), color=(c, cm))
Graphics3d Object
../../../_images/plot3d-4.svg

Beware that the color function must take values between 0 and 1.

We plot “cape man”:

sage: S = sphere(size=.5, color='yellow')

sage: from sage.plot.plot3d.shapes import Cone
sage: S += Cone(.5, .5, color='red').translate(0,0,.3)

sage: S += sphere((.45, -.1, .15), size=.1, color='white')
sage: S += sphere((.51,-.1,.17), size=.05, color='black')
sage: S += sphere((.45, .1, .15), size=.1, color='white')
sage: S += sphere((.51, .1,.17), size=.05, color='black')
sage: S += sphere((.5, 0, -.2), size=.1, color='yellow')
sage: def f(x,y): return math.exp(x/5)*math.cos(y)
sage: P = plot3d(f, (-5, 5), (-5, 5), adaptive=True,
....:            color=['red','yellow'], max_depth=10)
sage: cape_man = P.scale(.2) + S.translate(1, 0, 0)
sage: cape_man.show(aspect_ratio=[1, 1, 1])
../../../_images/plot3d-5.svg

Or, we plot a very simple function indeed:

sage: plot3d(pi, (-1,1), (-1,1))
Graphics3d Object
../../../_images/plot3d-6.svg

Transparent with fractional opacity value:

sage: plot3d(lambda x, y: x^2 + y^2, (-2,2), (-2,2), opacity=8/10)
Graphics3d Object

Todo

Add support for smooth triangles.

AUTHORS:

  • Tom Boothby: adaptive refinement triangles

  • Josh Kantor: adaptive refinement triangles

  • Robert Bradshaw (2007-08): initial version of this file

  • William Stein (2007-12, 2008-01): improving 3d plotting

  • Oscar Lazo, William Cauchois, Jason Grout (2009-2010): Adding coordinate transformations

class sage.plot.plot3d.plot3d.Cylindrical(dep_var, indep_vars)#

Bases: _Coordinates

A cylindrical coordinate system for use with plot3d(transformation=...) where the position of a point is specified by three numbers:

  • the radial distance (radius) from the \(z\)-axis

  • the azimuth angle (azimuth) from the positive \(x\)-axis

  • the height or altitude (height) above the \(xy\)-plane

These three variables must be specified in the constructor.

EXAMPLES:

Construct a cylindrical transformation for a function for height in terms of radius and azimuth:

sage: T = Cylindrical('height', ['radius', 'azimuth'])

If we construct some concrete variables, we can get a transformation:

sage: r, theta, z = var('r theta z')
sage: T.transform(radius=r, azimuth=theta, height=z)
(r*cos(theta), r*sin(theta), z)

We can plot with this transform. Remember that the dependent variable is the height, and the independent variables are the radius and the azimuth (in that order):

sage: plot3d(9-r^2, (r, 0, 3), (theta, 0, pi), transformation=T)
Graphics3d Object
../../../_images/plot3d-7.svg

We next graph the function where the radius is constant:

sage: S = Cylindrical('radius', ['azimuth', 'height'])
sage: theta, z = var('theta, z')
sage: plot3d(3, (theta, 0, 2*pi), (z, -2, 2), transformation=S)
Graphics3d Object
../../../_images/plot3d-8.svg

See also cylindrical_plot3d() for more examples of plotting in cylindrical coordinates.

transform(radius=None, azimuth=None, height=None)#

A cylindrical coordinates transform.

EXAMPLES:

sage: T = Cylindrical('height', ['azimuth', 'radius'])
sage: T.transform(radius=var('r'), azimuth=var('theta'), height=var('z'))
(r*cos(theta), r*sin(theta), z)
class sage.plot.plot3d.plot3d.Spherical(dep_var, indep_vars)#

Bases: _Coordinates

A spherical coordinate system for use with plot3d(transformation=...) where the position of a point is specified by three numbers:

  • the radial distance (radius) from the origin

  • the azimuth angle (azimuth) from the positive \(x\)-axis

  • the inclination angle (inclination) from the positive \(z\)-axis

These three variables must be specified in the constructor.

EXAMPLES:

Construct a spherical transformation for a function for the radius in terms of the azimuth and inclination:

sage: T = Spherical('radius', ['azimuth', 'inclination'])

If we construct some concrete variables, we can get a transformation in terms of those variables:

sage: r, phi, theta = var('r phi theta')
sage: T.transform(radius=r, azimuth=theta, inclination=phi)
(r*cos(theta)*sin(phi), r*sin(phi)*sin(theta), r*cos(phi))

We can plot with this transform. Remember that the dependent variable is the radius, and the independent variables are the azimuth and the inclination (in that order):

sage: plot3d(phi * theta, (theta, 0, pi), (phi, 0, 1), transformation=T)
Graphics3d Object
../../../_images/plot3d-9.svg

We next graph the function where the inclination angle is constant:

sage: S = Spherical('inclination', ['radius', 'azimuth'])
sage: r, theta = var('r,theta')
sage: plot3d(3, (r,0,3), (theta, 0, 2*pi), transformation=S)
Graphics3d Object
../../../_images/plot3d-10.svg

See also spherical_plot3d() for more examples of plotting in spherical coordinates.

transform(radius=None, azimuth=None, inclination=None)#

A spherical coordinates transform.

EXAMPLES:

sage: T = Spherical('radius', ['azimuth', 'inclination'])
sage: T.transform(radius=var('r'), azimuth=var('theta'), inclination=var('phi'))
(r*cos(theta)*sin(phi), r*sin(phi)*sin(theta), r*cos(phi))
class sage.plot.plot3d.plot3d.SphericalElevation(dep_var, indep_vars)#

Bases: _Coordinates

A spherical coordinate system for use with plot3d(transformation=...) where the position of a point is specified by three numbers:

  • the radial distance (radius) from the origin

  • the azimuth angle (azimuth) from the positive \(x\)-axis

  • the elevation angle (elevation) from the \(xy\)-plane toward the positive \(z\)-axis

These three variables must be specified in the constructor.

EXAMPLES:

Construct a spherical transformation for the radius in terms of the azimuth and elevation. Then, get a transformation in terms of those variables:

sage: T = SphericalElevation('radius', ['azimuth', 'elevation'])
sage: r, theta, phi = var('r theta phi')
sage: T.transform(radius=r, azimuth=theta, elevation=phi)
(r*cos(phi)*cos(theta), r*cos(phi)*sin(theta), r*sin(phi))

We can plot with this transform. Remember that the dependent variable is the radius, and the independent variables are the azimuth and the elevation (in that order):

sage: plot3d(phi * theta, (theta, 0, pi), (phi, 0, 1), transformation=T)
Graphics3d Object
../../../_images/plot3d-11.svg

We next graph the function where the elevation angle is constant. This should be compared to the similar example for the Spherical coordinate system:

sage: SE = SphericalElevation('elevation', ['radius', 'azimuth'])
sage: r, theta = var('r,theta')
sage: plot3d(3, (r, 0, 3), (theta, 0, 2*pi), transformation=SE)
Graphics3d Object
../../../_images/plot3d-12.svg

Plot a sin curve wrapped around the equator:

sage: P1 = plot3d((pi/12)*sin(8*theta), (r,0.99,1), (theta, 0, 2*pi),
....:             transformation=SE, plot_points=(10,200))
sage: P2 = sphere(center=(0,0,0), size=1, color='red', opacity=0.3)
sage: P1 + P2
Graphics3d Object
../../../_images/plot3d-13.svg

Now we graph several constant elevation functions alongside several constant inclination functions. This example illustrates the difference between the Spherical coordinate system and the SphericalElevation coordinate system:

sage: r, phi, theta = var('r phi theta')
sage: SE = SphericalElevation('elevation', ['radius', 'azimuth'])
sage: angles = [pi/18, pi/12, pi/6]
sage: P1 = [plot3d(a, (r,0,3), (theta, 0, 2*pi), transformation=SE,
....:              opacity=0.85, color='blue')
....:       for a in angles]

sage: S = Spherical('inclination', ['radius', 'azimuth'])
sage: P2 = [plot3d(a, (r,0,3), (theta, 0, 2*pi), transformation=S,
....:              opacity=0.85, color='red')
....:       for a in angles]
sage: show(sum(P1+P2), aspect_ratio=1)
../../../_images/plot3d-14.svg

See also spherical_plot3d() for more examples of plotting in spherical coordinates.

transform(radius=None, azimuth=None, elevation=None)#

A spherical elevation coordinates transform.

EXAMPLES:

sage: T = SphericalElevation('radius', ['azimuth', 'elevation'])
sage: T.transform(radius=var('r'), azimuth=var('theta'), elevation=var('phi'))
(r*cos(phi)*cos(theta), r*cos(phi)*sin(theta), r*sin(phi))
class sage.plot.plot3d.plot3d.TrivialTriangleFactory#

Bases: object

Class emulating behavior of TriangleFactory but simply returning a list of vertices for both regular and smooth triangles.

smooth_triangle(a, b, c, da, db, dc, color=None)#

Function emulating behavior of smooth_triangle() but simply returning a list of vertices.

INPUT:

  • a, b, c – triples (x,y,z) representing corners on a triangle in 3-space

  • da, db, dc – ignored

  • color – ignored

OUTPUT:

  • the list [a,b,c]

triangle(a, b, c, color=None)#

Function emulating behavior of triangle() but simply returning a list of vertices.

INPUT:

  • a, b, c – triples (x,y,z) representing corners on a triangle in 3-space

  • color – ignored

OUTPUT:

  • the list [a,b,c]

sage.plot.plot3d.plot3d.axes(scale=1, radius=None, **kwds)#

Creates basic axes in three dimensions. Each axis is a three dimensional arrow object.

INPUT:

  • scale – (default: 1) the length of the axes (all three will be the same)

  • radius – (default: .01) the radius of the axes as arrows

EXAMPLES:

sage: from sage.plot.plot3d.plot3d import axes
sage: S = axes(6, color='black'); S
Graphics3d Object
../../../_images/plot3d-15.svg
sage: T = axes(2, .5); T
Graphics3d Object
../../../_images/plot3d-16.svg
sage.plot.plot3d.plot3d.cylindrical_plot3d(f, urange, vrange, **kwds)#

Plots a function in cylindrical coordinates. This function is equivalent to:

sage: r, u, v = var('r,u,v')
sage: f = u*v; urange = (u, 0, pi); vrange = (v, 0, pi)
sage: T = (r*cos(u), r*sin(u), v, [u, v])
sage: plot3d(f, urange, vrange, transformation=T)
Graphics3d Object
../../../_images/plot3d-17.svg

or equivalently:

sage: T = Cylindrical('radius', ['azimuth', 'height'])
sage: f=lambda u,v: u*v; urange=(u,0,pi); vrange=(v,0,pi)
sage: plot3d(f, urange, vrange, transformation=T)
Graphics3d Object

INPUT:

  • f – a symbolic expression or function of two variables, representing the radius from the \(z\)-axis

  • urange – a 3-tuple (u, u_min, u_max), the domain of the azimuth variable

  • vrange – a 3-tuple (v, v_min, v_max), the domain of the elevation (\(z\)) variable

EXAMPLES:

A portion of a cylinder of radius 2:

sage: theta, z = var('theta,z')
sage: cylindrical_plot3d(2, (theta, 0, 3*pi/2), (z, -2, 2))
Graphics3d Object
../../../_images/plot3d-18.svg

Some random figures:

sage: cylindrical_plot3d(cosh(z), (theta, 0, 2*pi), (z, -2, 2))
Graphics3d Object
../../../_images/plot3d-19.svg
sage: cylindrical_plot3d(e^(-z^2)*(cos(4*theta) + 2) + 1, (theta, 0, 2*pi), (z, -2, 2), plot_points=[80, 80]).show(aspect_ratio=(1, 1, 1))
../../../_images/plot3d-20.svg
sage.plot.plot3d.plot3d.plot3d(f, urange, vrange, adaptive=False, transformation=None, **kwds)#

Plots a function in 3d.

INPUT:

  • f – a symbolic expression or function of 2 variables

  • urange – a 2-tuple (u_min, u_max) or a 3-tuple (u, u_min, u_max)

  • vrange – a 2-tuple (v_min, v_max) or a 3-tuple (v, v_min, v_max)

  • adaptive – (default: False) whether to use adaptive refinement to draw the plot (slower, but may look better). This option does NOT work in conjunction with a transformation (see below).

  • mesh – bool (default: False) whether to display mesh grid lines

  • dots – bool (default: False) whether to display dots at mesh grid points

  • plot_points – (default: “automatic”) initial number of sample points in each direction; an integer or a pair of integers

  • transformation – (default: None) a transformation to apply. May be a 3 or 4-tuple (x_func, y_func, z_func, independent_vars) where the first 3 items indicate a transformation to Cartesian coordinates (from your coordinate system) in terms of u, v, and the function variable fvar (for which the value of f will be substituted). If a 3-tuple is specified, the independent variables are chosen from the range variables. If a 4-tuple is specified, the 4th element is a list of independent variables. transformation may also be a predefined coordinate system transformation like Spherical or Cylindrical.

Note

mesh and dots are not supported when using the Tachyon raytracer renderer.

EXAMPLES: We plot a 3d function defined as a Python function:

sage: plot3d(lambda x, y: x^2 + y^2, (-2,2), (-2,2))
Graphics3d Object
../../../_images/plot3d-21.svg

We plot the same 3d function but using adaptive refinement:

sage: plot3d(lambda x, y: x^2 + y^2, (-2,2), (-2,2), adaptive=True)
Graphics3d Object
../../../_images/plot3d-22.svg

Adaptive refinement but with more points:

sage: plot3d(lambda x, y: x^2 + y^2, (-2,2), (-2,2), adaptive=True, initial_depth=5)
Graphics3d Object
../../../_images/plot3d-23.svg

We plot some 3d symbolic functions:

sage: var('x,y')
(x, y)
sage: plot3d(x^2 + y^2, (x,-2,2), (y,-2,2))
Graphics3d Object
../../../_images/plot3d-24.svg
sage: plot3d(sin(x*y), (x, -pi, pi), (y, -pi, pi))
Graphics3d Object
../../../_images/plot3d-25.svg

We give a plot with extra sample points:

sage: var('x,y')
(x, y)
sage: plot3d(sin(x^2 + y^2), (x,-5,5), (y,-5,5), plot_points=200)
Graphics3d Object
../../../_images/plot3d-26.svg
sage: plot3d(sin(x^2 + y^2), (x, -5, 5), (y, -5, 5), plot_points=[10, 100])
Graphics3d Object
../../../_images/plot3d-27.svg

A 3d plot with a mesh:

sage: var('x,y')
(x, y)
sage: plot3d(sin(x - y)*y*cos(x), (x, -3, 3), (y, -3, 3), mesh=True)
Graphics3d Object
../../../_images/plot3d-28.svg

The same with thicker mesh lines (not supported in all viewers):

sage: var('x,y')
(x, y)
sage: plot3d(sin(x - y)*y*cos(x), (x, -3, 3), (y, -3, 3), mesh=True,
....:        thickness=2, viewer='threejs')
Graphics3d Object

Two wobby translucent planes:

sage: x,y = var('x,y')
sage: P = plot3d(x + y + sin(x*y), (x, -10, 10), (y, -10, 10),
....:            opacity=0.87, color='blue')
sage: Q = plot3d(x - 2*y - cos(x*y),(x, -10, 10), (y, -10, 10),
....:            opacity=0.3, color='red')
sage: P + Q
Graphics3d Object
../../../_images/plot3d-29.svg

We draw two parametric surfaces and a transparent plane:

sage: L = plot3d(lambda x,y: 0, (-5,5), (-5,5), color="lightblue", opacity=0.8)
sage: P = plot3d(lambda x,y: 4 - x^3 - y^2, (-2,2), (-2,2), color='green')
sage: Q = plot3d(lambda x,y: x^3 + y^2 - 4, (-2,2), (-2,2), color='orange')
sage: L + P + Q
Graphics3d Object
../../../_images/plot3d-30.svg

We draw the “Sinus” function (water ripple-like surface):

sage: x, y = var('x y')
sage: plot3d(sin(pi*(x^2 + y^2))/2, (x, -1, 1), (y, -1, 1))
Graphics3d Object
../../../_images/plot3d-31.svg

Hill and valley (flat surface with a bump and a dent):

sage: x, y = var('x y')
sage: plot3d(4*x*exp(-x^2 - y^2), (x, -2, 2), (y, -2, 2))
Graphics3d Object
../../../_images/plot3d-32.svg

An example of a transformation:

sage: r, phi, z = var('r phi z')
sage: trans = (r*cos(phi), r*sin(phi), z)
sage: plot3d(cos(r), (r, 0, 17*pi/2), (phi, 0, 2*pi), transformation=trans, opacity=0.87).show(aspect_ratio=(1,1,2), frame=False)
../../../_images/plot3d-33.svg

An example of a transformation with symbolic vector:

sage: cylindrical(r, theta, z) = [r*cos(theta), r*sin(theta), z]
sage: plot3d(3, (theta, 0, pi/2), (z, 0, pi/2), transformation=cylindrical)
Graphics3d Object
../../../_images/plot3d-34.svg

Many more examples of transformations:

sage: u, v, w = var('u v w')
sage: rectangular=(u,v,w)
sage: spherical=(w*cos(u)*sin(v),w*sin(u)*sin(v),w*cos(v))
sage: cylindric_radial=(w*cos(u),w*sin(u),v)
sage: cylindric_axial=(v*cos(u),v*sin(u),w)
sage: parabolic_cylindrical=(w*v,(v^2-w^2)/2,u)

Plot a constant function of each of these to get an idea of what it does:

sage: A = plot3d(2,(u,-pi,pi),(v,0,pi),transformation=rectangular,plot_points=[100,100])
sage: B = plot3d(2,(u,-pi,pi),(v,0,pi),transformation=spherical,plot_points=[100,100])
sage: C = plot3d(2,(u,-pi,pi),(v,0,pi),transformation=cylindric_radial,plot_points=[100,100])
sage: D = plot3d(2,(u,-pi,pi),(v,0,pi),transformation=cylindric_axial,plot_points=[100,100])
sage: E = plot3d(2,(u,-pi,pi),(v,-pi,pi),transformation=parabolic_cylindrical,plot_points=[100,100])
sage: @interact
....: def _(which_plot=[A,B,C,D,E]):
....:     show(which_plot)
...Interactive function <function _ at ...> with 1 widget
  which_plot: Dropdown(description='which_plot', options=(Graphics3d Object, Graphics3d Object, Graphics3d Object, Graphics3d Object, Graphics3d Object), value=Graphics3d Object)

Now plot a function:

sage: g=3+sin(4*u)/2+cos(4*v)/2
sage: F = plot3d(g,(u,-pi,pi),(v,0,pi),transformation=rectangular,plot_points=[100,100])
sage: G = plot3d(g,(u,-pi,pi),(v,0,pi),transformation=spherical,plot_points=[100,100])
sage: H = plot3d(g,(u,-pi,pi),(v,0,pi),transformation=cylindric_radial,plot_points=[100,100])
sage: I = plot3d(g,(u,-pi,pi),(v,0,pi),transformation=cylindric_axial,plot_points=[100,100])
sage: J = plot3d(g,(u,-pi,pi),(v,0,pi),transformation=parabolic_cylindrical,plot_points=[100,100])
sage: @interact
....: def _(which_plot=[F, G, H, I, J]):
....:     show(which_plot)
...Interactive function <function _ at ...> with 1 widget
  which_plot: Dropdown(description='which_plot', options=(Graphics3d Object, Graphics3d Object, Graphics3d Object, Graphics3d Object, Graphics3d Object), value=Graphics3d Object)
sage.plot.plot3d.plot3d.plot3d_adaptive(f, x_range, y_range, color='automatic', grad_f=None, max_bend=0.5, max_depth=5, initial_depth=4, num_colors=128, **kwds)#

Adaptive 3d plotting of a function of two variables.

This is used internally by the plot3d command when the option adaptive=True is given.

INPUT:

  • f – a symbolic function or a Python function of 3 variables

  • x_range – x range of values: 2-tuple (xmin,

    xmax) or 3-tuple (x,xmin,xmax)

  • y_range – y range of values: 2-tuple (ymin, ymax) or 3-tuple

    (y,ymin,ymax)

  • grad_f – gradient of f as a Python function

  • color – “automatic” - a rainbow of num_colors colors

  • num_colors – (default: 128) number of colors to use with default

    color

  • max_bend – (default: 0.5)

  • max_depth – (default: 5)

  • initial_depth – (default: 4)

  • **kwds – standard graphics parameters

EXAMPLES:

We plot \(\sin(xy)\):

sage: from sage.plot.plot3d.plot3d import plot3d_adaptive
sage: x, y = var('x,y')
sage: plot3d_adaptive(sin(x*y), (x, -pi, pi), (y, -pi, pi), initial_depth=5)
Graphics3d Object
../../../_images/plot3d-35.svg
sage.plot.plot3d.plot3d.spherical_plot3d(f, urange, vrange, **kwds)#

Plots a function in spherical coordinates. This function is equivalent to:

sage: r,u,v=var('r,u,v')
sage: f=u*v; urange=(u,0,pi); vrange=(v,0,pi)
sage: T = (r*cos(u)*sin(v), r*sin(u)*sin(v), r*cos(v), [u,v])
sage: plot3d(f, urange, vrange, transformation=T)
Graphics3d Object

or equivalently:

sage: T = Spherical('radius', ['azimuth', 'inclination'])
sage: f=lambda u,v: u*v; urange=(u,0,pi); vrange=(v,0,pi)
sage: plot3d(f, urange, vrange, transformation=T)
Graphics3d Object

INPUT:

  • f – a symbolic expression or function of two variables

  • urange – a 3-tuple (u, u_min, u_max), the domain of the azimuth variable

  • vrange – a 3-tuple (v, v_min, v_max), the domain of the inclination variable

EXAMPLES:

A sphere of radius 2:

sage: x,y = var('x,y')
sage: spherical_plot3d(2, (x, 0, 2*pi), (y, 0, pi))
Graphics3d Object
../../../_images/plot3d-36.svg

The real and imaginary parts of a spherical harmonic with \(l=2\) and \(m=1\):

sage: phi, theta = var('phi, theta')
sage: Y = spherical_harmonic(2, 1, theta, phi)
sage: rea = spherical_plot3d(abs(real(Y)), (phi, 0, 2*pi), (theta, 0, pi), color='blue', opacity=0.6)
sage: ima = spherical_plot3d(abs(imag(Y)), (phi, 0, 2*pi), (theta, 0, pi), color='red', opacity=0.6)
sage: (rea + ima).show(aspect_ratio=1)  # long time (4s on sage.math, 2011)
../../../_images/plot3d-37.svg

A drop of water:

sage: x,y = var('x,y')
sage: spherical_plot3d(e^-y, (x, 0, 2*pi), (y, 0, pi), opacity=0.5).show(frame=False)
../../../_images/plot3d-38.svg

An object similar to a heart:

sage: x,y = var('x,y')
sage: spherical_plot3d((2 + cos(2*x))*(y + 1), (x, 0, 2*pi), (y, 0, pi), rgbcolor=(1, .1, .1))
Graphics3d Object
../../../_images/plot3d-39.svg

Some random figures:

sage: x,y = var('x,y')
sage: spherical_plot3d(1 + sin(5*x)/5, (x, 0, 2*pi), (y, 0, pi), rgbcolor=(1, 0.5, 0), plot_points=(80, 80), opacity=0.7)
Graphics3d Object
../../../_images/plot3d-40.svg
sage: x, y = var('x,y')
sage: spherical_plot3d(1 + 2*cos(2*y), (x, 0, 3*pi/2), (y, 0, pi)).show(aspect_ratio=(1, 1, 1))
../../../_images/plot3d-41.svg