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# Plotting Functions

EXAMPLES:

    sage: def f(x,y):
    ...       return math.sin(y*y+x*x)/math.sqrt(x*x+y*y+.0001)
    ...
    sage: P = plot3d(f,(-3,3),(-3,3), adaptive=True, color=rainbow(60, 'rgbtuple'), max_bend=.1, max_depth=15)
    sage: P.show()

    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()

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') + sphere((.51,-.1,.17), size=.05, color='black')
    sage: S += sphere((.45, .1,.15),size=.1, color='white') + 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])

Or, we plot a very simple function indeed:

    sage: plot3d(pi, (-1,1), (-1,1))
    Graphics3d Object

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: `sage.plot.plot3d.plot3d._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

  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

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

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

    A cylindrical coordinates transform.

    EXAMPLE:

        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: `sage.plot.plot3d.plot3d._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

  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

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

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

    A spherical coordinates transform.

    EXAMPLE:

        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: `sage.plot.plot3d.plot3d._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

  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

  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

  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)

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

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

    A spherical elevation coordinates transform.

    EXAMPLE:

        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`

  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]`

    TESTS:

        sage: from sage.plot.plot3d.plot3d import TrivialTriangleFactory
        sage: factory = TrivialTriangleFactory()
        sage: sm_tri = factory.smooth_triangle([0,0,0],[0,0,1],[1,1,0],[0,0,1],[0,2,0],[1,0,0])
        sage: sm_tri
        [[0, 0, 0], [0, 0, 1], [1, 1, 0]]

  `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]`

    TESTS:

        sage: from sage.plot.plot3d.plot3d import TrivialTriangleFactory
        sage: factory = TrivialTriangleFactory()
        sage: tri = factory.triangle([0,0,0],[0,0,1],[1,1,0])
        sage: tri
        [[0, 0, 0], [0, 0, 1], [1, 1, 0]]

`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

      sage: T = axes(2, .5); T
      Graphics3d Object

`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

  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

  Some random figures:

      sage: cylindrical_plot3d(cosh(z),(theta,0,2*pi),(z,-2,2))
      Graphics3d Object

      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))

`sage.plot.plot3d.plot3d.``plot3d`(f, urange, vrange, adaptive=False, transformation=None, **kwds)

  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 conjuction 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

  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

  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

  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
      sage: plot3d(sin(x*y), (x, -pi, pi), (y, -pi, pi))
      Graphics3d Object

  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
      sage: plot3d(sin(x^2+y^2),(x,-5,5),(y,-5,5), plot_points=[10,100])
      Graphics3d Object

  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

  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

  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

  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

  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

  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)

  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

  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)
      <html>...

  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)
      <html>...

  TESTS:

  Make sure the transformation plots work:

      sage: show(A + B + C + D + E)
      sage: show(F + G + H + I + J)

  Listing the same plot variable twice gives an error:

      sage: x, y = var('x y')
      sage: plot3d( 4*x*exp(-x^2-y^2), (x,-2,2), (x,-2,2))
      Traceback (most recent call last):
      ...
      ValueError: range variables should be distinct, but there are duplicates

`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'); plot3d_adaptive(sin(x*y), (x,-pi,pi), (y,-pi,pi), initial_depth=5)
      Graphics3d Object

`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

  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)

  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)

  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

  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

      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))

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