CoCalc -- parametric

GitHub Repository: sagemath/sagelib
Path: blob/master/sage/plot/plot3d/parametric_plot3d.py
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"""
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Parametric Plots
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"""
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from parametric_surface import ParametricSurface
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from shapes2 import line3d
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from sage.misc.misc import xsrange, srange
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from sage.structure.element import is_Vector
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from sage.ext.fast_eval import fast_float, fast_float_constant
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def parametric_plot3d(f, urange, vrange=None, plot_points="automatic", boundary_style=None, **kwds):
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    r"""
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    Return a parametric three-dimensional space curve or surface.
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    There are four ways to call this function:
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    - ``parametric_plot3d([f_x, f_y, f_z], (u_min,
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      u_max))``:
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      `f_x, f_y, f_z` are three functions and
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      `u_{\min}` and `u_{\max}` are real numbers
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    - ``parametric_plot3d([f_x, f_y, f_z], (u, u_min,
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      u_max))``:
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      `f_x, f_y, f_z` can be viewed as functions of
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      `u`
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    - ``parametric_plot3d([f_x, f_y, f_z], (u_min, u_max),
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      (v_min, v_max))``:
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      `f_x, f_y, f_z` are each functions of two variables
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    - ``parametric_plot3d([f_x, f_y, f_z], (u, u_min,
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      u_max), (v, v_min, v_max))``:
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      `f_x, f_y, f_z` can be viewed as functions of
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      `u` and `v`
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    INPUT:
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    - ``f`` - a 3-tuple of functions or expressions, or vector of size 3
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    - ``urange`` - a 2-tuple (u_min, u_max) or a 3-tuple
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      (u, u_min, u_max)
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    - ``vrange`` - (optional - only used for surfaces) a
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      2-tuple (v_min, v_max) or a 3-tuple (v, v_min, v_max)
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    - ``plot_points`` - (default: "automatic", which is
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      75 for curves and [40,40] for surfaces) initial number of sample
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      points in each parameter; an integer for a curve, and a pair of
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      integers for a surface.
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    - ``boundary_style`` - (default: None, no boundary) a dict that describes
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      how to draw the boundaries of regions by giving options that are passed
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      to the line3d command.
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    - ``mesh`` - bool (default: False) whether to display
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      mesh grid lines
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    - ``dots`` - bool (default: False) whether to display
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      dots at mesh grid points
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    .. note::
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       #. By default for a curve any points where `f_x`,
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          `f_y`, or `f_z` do not evaluate to a real number
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          are skipped.
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       #. Currently for a surface `f_x`, `f_y`, and
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          `f_z` have to be defined everywhere. This will change.
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       #. mesh and dots are not supported when using the Tachyon ray tracer
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          renderer.
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    EXAMPLES: We demonstrate each of the four ways to call this
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    function.
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    #. A space curve defined by three functions of 1 variable:
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       ::
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           sage: parametric_plot3d( (sin, cos, lambda u: u/10), (0, 20))
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       Note above the lambda function, which creates a callable Python
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       function that sends `u` to `u/10`.
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    #. Next we draw the same plot as above, but using symbolic
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       functions:
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       ::
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           sage: u = var('u')
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           sage: parametric_plot3d( (sin(u), cos(u), u/10), (u, 0, 20))
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    #. We draw a parametric surface using 3 Python functions (defined
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       using lambda):
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       ::
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           sage: f = (lambda u,v: cos(u), lambda u,v: sin(u)+cos(v), lambda u,v: sin(v))
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           sage: parametric_plot3d(f, (0, 2*pi), (-pi, pi))
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    #. The surface, but with a mesh:
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       ::
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           sage: u, v = var('u,v')
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           sage: parametric_plot3d((cos(u), sin(u) + cos(v), sin(v)), (u, 0, 2*pi), (v, -pi, pi), mesh=True)        
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    #. The same surface, but where the defining functions are
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       symbolic:
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       ::
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           sage: u, v = var('u,v')
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           sage: parametric_plot3d((cos(u), sin(u) + cos(v), sin(v)), (u, 0, 2*pi), (v, -pi, pi))
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       We increase the number of plot points, and make the surface green
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       and transparent:
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       ::
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           sage: parametric_plot3d((cos(u), sin(u) + cos(v), sin(v)), (u, 0, 2*pi), (v, -pi, pi), color='green', opacity=0.1, plot_points=[30,30])
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    We call the space curve function but with polynomials instead of
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    symbolic variables.
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    ::
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        sage: R.<t> = RDF[]
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        sage: parametric_plot3d( (t, t^2, t^3), (t, 0, 3) ) 
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    Next we plot the same curve, but because we use (0, 3) instead of
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    (t, 0, 3), each polynomial is viewed as a callable function of one
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    variable::
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        sage: parametric_plot3d( (t, t^2, t^3), (0, 3) ) 
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    We do a plot but mix a symbolic input, and an integer::
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        sage: t = var('t')
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        sage: parametric_plot3d( (1, sin(t), cos(t)), (t, 0, 3) ) 
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    We specify a boundary style to show us the values of the function at its
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    extrema::
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        sage: u, v = var('u,v')
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        sage: parametric_plot3d((cos(u), sin(u) + cos(v), sin(v)), (u, 0, pi), (v, 0, pi), \
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        ...                     boundary_style={"color": "black", "thickness": 2})
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    We can plot vectors::
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        sage: x,y=var('x,y')
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        sage: parametric_plot3d(vector([x-y,x*y,x*cos(y)]), (x,0,2), (y,0,2))
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        sage: t=var('t')
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        sage: p=vector([1,2,3])
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        sage: q=vector([2,-1,2])
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        sage: parametric_plot3d(p*t+q, (t, 0, 2))
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    Any options you would normally use to specify the appearance of a curve are
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    valid as entries in the boundary_style dict.
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    MANY MORE EXAMPLES:
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    We plot two interlinked tori::
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        sage: u, v = var('u,v')
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        sage: f1 = (4+(3+cos(v))*sin(u), 4+(3+cos(v))*cos(u), 4+sin(v))
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        sage: f2 = (8+(3+cos(v))*cos(u), 3+sin(v), 4+(3+cos(v))*sin(u))
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        sage: p1 = parametric_plot3d(f1, (u,0,2*pi), (v,0,2*pi), texture="red")
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        sage: p2 = parametric_plot3d(f2, (u,0,2*pi), (v,0,2*pi), texture="blue")
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        sage: p1 + p2
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    A cylindrical Star of David::
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        sage: u,v = var('u v')
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        sage: f_x = cos(u)*cos(v)*(abs(cos(3*v/4))^500 + abs(sin(3*v/4))^500)^(-1/260)*(abs(cos(4*u/4))^200 + abs(sin(4*u/4))^200)^(-1/200)
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        sage: f_y = cos(u)*sin(v)*(abs(cos(3*v/4))^500 + abs(sin(3*v/4))^500)^(-1/260)*(abs(cos(4*u/4))^200 + abs(sin(4*u/4))^200)^(-1/200)
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        sage: f_z = sin(u)*(abs(cos(4*u/4))^200 + abs(sin(4*u/4))^200)^(-1/200)
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        sage: parametric_plot3d([f_x, f_y, f_z], (u, -pi, pi), (v, 0, 2*pi))
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    Double heart::
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        sage: u, v = var('u,v')
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        sage: f_x = ( abs(v) - abs(u) - abs(tanh((1/sqrt(2))*u)/(1/sqrt(2))) + abs(tanh((1/sqrt(2))*v)/(1/sqrt(2))) )*sin(v)
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        sage: f_y = ( abs(v) - abs(u) - abs(tanh((1/sqrt(2))*u)/(1/sqrt(2))) - abs(tanh((1/sqrt(2))*v)/(1/sqrt(2))) )*cos(v)
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        sage: f_z = sin(u)*(abs(cos(4*u/4))^1 + abs(sin(4*u/4))^1)^(-1/1)
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        sage: parametric_plot3d([f_x, f_y, f_z], (u, 0, pi), (v, -pi, pi))
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    Heart::
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        sage: u, v = var('u,v')
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        sage: f_x = cos(u)*(4*sqrt(1-v^2)*sin(abs(u))^abs(u))
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        sage: f_y = sin(u) *(4*sqrt(1-v^2)*sin(abs(u))^abs(u))
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        sage: f_z = v
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        sage: parametric_plot3d([f_x, f_y, f_z], (u, -pi, pi), (v, -1, 1), frame=False, color="red")
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    Green bowtie::
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        sage: u, v = var('u,v')
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        sage: f_x = sin(u) / (sqrt(2) + sin(v))
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        sage: f_y = sin(u) / (sqrt(2) + cos(v))
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        sage: f_z = cos(u) / (1 + sqrt(2))
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        sage: parametric_plot3d([f_x, f_y, f_z], (u, -pi, pi), (v, -pi, pi), frame=False, color="green")
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    Boy's surface http://en.wikipedia.org/wiki/Boy's_surface
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    ::
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        sage: u, v = var('u,v')
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        sage: fx = 2/3* (cos(u)* cos(2*v) + sqrt(2)* sin(u)* cos(v))* cos(u) / (sqrt(2) - sin(2*u)* sin(3*v))
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        sage: fy = 2/3* (cos(u)* sin(2*v) - sqrt(2)* sin(u)* sin(v))* cos(u) / (sqrt(2) - sin(2*u)* sin(3*v))
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        sage: fz = sqrt(2)* cos(u)* cos(u) / (sqrt(2) - sin(2*u)* sin(3*v))
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        sage: parametric_plot3d([fx, fy, fz], (u, -2*pi, 2*pi), (v, 0, pi), plot_points = [90,90], frame=False, color="orange") # long time -- about 30 seconds
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    Maeder's_Owl (pretty but can't find an internet reference)::
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        sage: u, v = var('u,v')
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        sage: fx = v *cos(u) - 0.5* v^2 * cos(2* u)
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        sage: fy = -v *sin(u) - 0.5* v^2 * sin(2* u)
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        sage: fz = 4 *v^1.5 * cos(3 *u / 2) / 3
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        sage: parametric_plot3d([fx, fy, fz], (u, -2*pi, 2*pi), (v, 0, 1),plot_points = [90,90], frame=False, color="purple")
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    Bracelet::
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        sage: u, v = var('u,v')
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        sage: fx = (2 + 0.2*sin(2*pi*u))*sin(pi*v)
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        sage: fy = 0.2*cos(2*pi*u) *3*cos(2*pi*v)
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        sage: fz = (2 + 0.2*sin(2*pi*u))*cos(pi*v)
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        sage: parametric_plot3d([fx, fy, fz], (u, 0, pi/2), (v, 0, 3*pi/4), frame=False, color="gray")
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    Green goblet
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    ::
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        sage: u, v = var('u,v')
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        sage: fx = cos(u)*cos(2*v)
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        sage: fy = sin(u)*cos(2*v)
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        sage: fz = sin(v)
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        sage: parametric_plot3d([fx, fy, fz], (u, 0, 2*pi), (v, 0, pi), frame=False, color="green")
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    Funny folded surface - with square projection::
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        sage: u, v = var('u,v')
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        sage: fx = cos(u)*sin(2*v)
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        sage: fy = sin(u)*cos(2*v)
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        sage: fz = sin(v)
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        sage: parametric_plot3d([fx, fy, fz], (u, 0, 2*pi), (v, 0, 2*pi), frame=False, color="green")
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    Surface of revolution of figure 8::
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        sage: u, v = var('u,v')
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        sage: fx = cos(u)*sin(2*v)
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        sage: fy = sin(u)*sin(2*v)
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        sage: fz = sin(v)
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        sage: parametric_plot3d([fx, fy, fz], (u, 0, 2*pi), (v, 0, 2*pi), frame=False, color="green")
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    Yellow Whitney's umbrella
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    http://en.wikipedia.org/wiki/Whitney_umbrella::
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        sage: u, v = var('u,v')
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        sage: fx = u*v
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        sage: fy = u
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        sage: fz = v^2
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        sage: parametric_plot3d([fx, fy, fz], (u, -1, 1), (v, -1, 1), frame=False, color="yellow")
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    Cross cap http://en.wikipedia.org/wiki/Cross-cap::
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        sage: u, v = var('u,v')
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        sage: fx = (1+cos(v))*cos(u)
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        sage: fy = (1+cos(v))*sin(u)
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        sage: fz = -tanh((2/3)*(u-pi))*sin(v)
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        sage: parametric_plot3d([fx, fy, fz], (u, 0, 2*pi), (v, 0, 2*pi), frame=False, color="red")
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    Twisted torus::
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        sage: u, v = var('u,v')
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        sage: fx = (3+sin(v)+cos(u))*cos(2*v)
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        sage: fy = (3+sin(v)+cos(u))*sin(2*v)
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        sage: fz = sin(u)+2*cos(v)
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        sage: parametric_plot3d([fx, fy, fz], (u, 0, 2*pi), (v, 0, 2*pi), frame=False, color="red")
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    Four intersecting discs::
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        sage: u, v = var('u,v')
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        sage: fx = v *cos(u) -0.5*v^2*cos(2*u)
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        sage: fy = -v*sin(u) -0.5*v^2*sin(2*u)
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        sage: fz = 4* v^1.5 *cos(3* u / 2) / 3
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        sage: parametric_plot3d([fx, fy, fz], (u, 0, 4*pi), (v, 0,2*pi), frame=False, color="red", opacity=0.7)
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    Steiner surface/Roman's surface (see
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    http://en.wikipedia.org/wiki/Roman_surface and
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    http://en.wikipedia.org/wiki/Steiner_surface)::
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        sage: u, v = var('u,v')
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        sage: fx = (sin(2 * u) * cos(v) * cos(v))
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        sage: fy = (sin(u) * sin(2 * v))
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        sage: fz = (cos(u) * sin(2 * v))
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        sage: parametric_plot3d([fx, fy, fz], (u, -pi/2, pi/2), (v, -pi/2,pi/2), frame=False, color="red")
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    Klein bottle? (see http://en.wikipedia.org/wiki/Klein_bottle)::
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        sage: u, v = var('u,v')
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        sage: fx = (3*(1+sin(v)) + 2*(1-cos(v)/2)*cos(u))*cos(v)
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        sage: fy = (4+2*(1-cos(v)/2)*cos(u))*sin(v)
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        sage: fz = -2*(1-cos(v)/2) * sin(u)
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        sage: parametric_plot3d([fx, fy, fz], (u, 0, 2*pi), (v, 0, 2*pi), frame=False, color="green")
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    A Figure 8 embedding of the Klein bottle (see
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    http://en.wikipedia.org/wiki/Klein_bottle)::
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        sage: u, v = var('u,v')
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        sage: fx = (2 + cos(v/2)* sin(u) - sin(v/2)* sin(2 *u))* cos(v)
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        sage: fy = (2 + cos(v/2)* sin(u) - sin(v/2)* sin(2 *u))* sin(v)
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        sage: fz = sin(v/2)* sin(u) + cos(v/2) *sin(2* u)
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        sage: parametric_plot3d([fx, fy, fz], (u, 0, 2*pi), (v, 0, 2*pi), frame=False, color="red")
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    Enneper's surface (see
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    http://en.wikipedia.org/wiki/Enneper_surface)::
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        sage: u, v = var('u,v')
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        sage: fx = u -u^3/3  + u*v^2
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        sage: fy = v -v^3/3  + v*u^2
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        sage: fz = u^2 - v^2
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        sage: parametric_plot3d([fx, fy, fz], (u, -2, 2), (v, -2, 2), frame=False, color="red")
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    Henneberg's surface (see
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    http://xahlee.org/surface/gallery_m.html)
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    ::
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        sage: u, v = var('u,v')
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        sage: fx = 2*sinh(u)*cos(v) -(2/3)*sinh(3*u)*cos(3*v)
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        sage: fy = 2*sinh(u)*sin(v) +(2/3)*sinh(3*u)*sin(3*v)
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        sage: fz = 2*cosh(2*u)*cos(2*v)
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        sage: parametric_plot3d([fx, fy, fz], (u, -1, 1), (v, -pi/2, pi/2), frame=False, color="red")
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    Dini's spiral
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    ::
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        sage: u, v = var('u,v')
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        sage: fx = cos(u)*sin(v)
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        sage: fy = sin(u)*sin(v)
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        sage: fz = (cos(v)+log(tan(v/2))) + 0.2*u
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        sage: parametric_plot3d([fx, fy, fz], (u, 0, 12.4), (v, 0.1, 2),frame=False, color="red")
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    Catalan's surface (see
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    http://xahlee.org/surface/catalan/catalan.html)::
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        sage: u, v = var('u,v')
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        sage: fx = u-sin(u)*cosh(v)
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        sage: fy = 1-cos(u)*cosh(v)
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        sage: fz = 4*sin(1/2*u)*sinh(v/2)
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        sage: parametric_plot3d([fx, fy, fz], (u, -pi, 3*pi), (v, -2, 2), frame=False, color="red")        
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    A Conchoid::
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        sage: u, v = var('u,v')
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        sage: k = 1.2; k_2 = 1.2; a = 1.5
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        sage: f = (k^u*(1+cos(v))*cos(u), k^u*(1+cos(v))*sin(u), k^u*sin(v)-a*k_2^u)
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        sage: parametric_plot3d(f, (u,0,6*pi), (v,0,2*pi), plot_points=[40,40], texture=(0,0.5,0))
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    A Mobius strip::
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        sage: u,v = var("u,v")
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        sage: parametric_plot3d([cos(u)*(1+v*cos(u/2)), sin(u)*(1+v*cos(u/2)), 0.2*v*sin(u/2)], (u,0, 4*pi+0.5), (v,0, 0.3),plot_points=[50,50])
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    A Twisted Ribbon
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    ::
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        sage: u, v = var('u,v')
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        sage: parametric_plot3d([3*sin(u)*cos(v), 3*sin(u)*sin(v), cos(v)], (u,0, 2*pi), (v, 0, pi),plot_points=[50,50])
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    An Ellipsoid::
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        sage: u, v = var('u,v')
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        sage: parametric_plot3d([3*sin(u)*cos(v), 2*sin(u)*sin(v), cos(u)], (u,0, 2*pi), (v, 0, 2*pi),plot_points=[50,50], aspect_ratio=[1,1,1])
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    A Cone::
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        sage: u, v = var('u,v')
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        sage: parametric_plot3d([u*cos(v), u*sin(v), u], (u, -1, 1), (v, 0, 2*pi+0.5), plot_points=[50,50])
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    A Paraboloid::
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        sage: u, v = var('u,v')
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        sage: parametric_plot3d([u*cos(v), u*sin(v), u^2], (u, 0, 1), (v, 0, 2*pi+0.4), plot_points=[50,50])
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    A Hyperboloid::
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        sage: u, v = var('u,v')
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        sage: plot3d(u^2-v^2, (u, -1, 1), (v, -1, 1), plot_points=[50,50])
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    A weird looking surface - like a Mobius band but also an O::
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        sage: u, v = var('u,v')
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        sage: parametric_plot3d([sin(u)*cos(u)*log(u^2)*sin(v), (u^2)^(1/6)*(cos(u)^2)^(1/4)*cos(v), sin(v)], (u, 0.001, 1), (v, -pi, pi+0.2), plot_points=[50,50])
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    A heart, but not a cardioid (for my wife)::
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        sage: u, v = var('u,v')
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        sage: p1 = parametric_plot3d([sin(u)*cos(u)*log(u^2)*v*(1-v)/2, ((u^6)^(1/20)*(cos(u)^2)^(1/4)-1/2)*v*(1-v), v^(0.5)], (u, 0.001, 1), (v, 0, 1), plot_points=[70,70], color='red')
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        sage: p2 = parametric_plot3d([-sin(u)*cos(u)*log(u^2)*v*(1-v)/2, ((u^6)^(1/20)*(cos(u)^2)^(1/4)-1/2)*v*(1-v), v^(0.5)], (u, 0.001, 1), (v, 0, 1), plot_points=[70,70], color='red')
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        sage: show(p1+p2, frame=False)
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    A Hyperhelicoidal::
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        sage: u = var("u")
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        sage: v = var("v")
416
        sage: fx = (sinh(v)*cos(3*u))/(1+cosh(u)*cosh(v))
417
        sage: fy = (sinh(v)*sin(3*u))/(1+cosh(u)*cosh(v))
418
        sage: fz = (cosh(v)*sinh(u))/(1+cosh(u)*cosh(v))
419
        sage: parametric_plot3d([fx, fy, fz], (u, -pi, pi), (v, -pi, pi), plot_points = [50,50], frame=False, color="red")
420
    
421
    A Helicoid (lines through a helix,
422
    http://en.wikipedia.org/wiki/Helix)::
423
    
424
        sage: u, v = var('u,v')
425
        sage: fx = sinh(v)*sin(u)
426
        sage: fy = -sinh(v)*cos(u)
427
        sage: fz = 3*u
428
        sage: parametric_plot3d([fx, fy, fz], (u, -pi, pi), (v, -pi, pi), plot_points = [50,50], frame=False, color="red")
429
    
430
    Kuen's surface
431
    (http://www.math.umd.edu/research/bianchi/Gifccsurfs/ccsurfs.html)::
432
    
433
        sage: fx = (2*(cos(u) + u*sin(u))*sin(v))/(1+ u^2*sin(v)^2)
434
        sage: fy = (2*(sin(u) - u*cos(u))*sin(v))/(1+ u^2*sin(v)^2)
435
        sage: fz = log(tan(1/2 *v)) + (2*cos(v))/(1+ u^2*sin(v)^2)
436
        sage: parametric_plot3d([fx, fy, fz], (u, 0, 2*pi), (v, 0.01, pi-0.01), plot_points = [50,50], frame=False, color="green")
437
    
438
    A 5-pointed star::
439
    
440
        sage: fx = cos(u)*cos(v)*(abs(cos(1*u/4))^0.5 + abs(sin(1*u/4))^0.5)^(-1/0.3)*(abs(cos(5*v/4))^1.7 + abs(sin(5*v/4))^1.7)^(-1/0.1)
441
        sage: fy = cos(u)*sin(v)*(abs(cos(1*u/4))^0.5 + abs(sin(1*u/4))^0.5)^(-1/0.3)*(abs(cos(5*v/4))^1.7 + abs(sin(5*v/4))^1.7)^(-1/0.1)
442
        sage: fz = sin(u)*(abs(cos(1*u/4))^0.5 + abs(sin(1*u/4))^0.5)^(-1/0.3)
443
        sage: parametric_plot3d([fx, fy, fz], (u, -pi/2, pi/2), (v, 0, 2*pi), plot_points = [50,50], frame=False, color="green")
444
    
445
    A cool self-intersecting surface (Eppener surface?)::
446
    
447
        sage: fx = u - u^3/3 + u*v^2
448
        sage: fy = v - v^3/3 + v*u^2
449
        sage: fz = u^2 - v^2
450
        sage: parametric_plot3d([fx, fy, fz], (u, -25, 25), (v, -25, 25), plot_points = [50,50], frame=False, color="green")
451
    
452
    The breather surface
453
    (http://en.wikipedia.org/wiki/Breather_surface)::
454
    
455
        sage: fx = (2*sqrt(0.84)*cosh(0.4*u)*(-(sqrt(0.84)*cos(v)*cos(sqrt(0.84)*v)) - sin(v)*sin(sqrt(0.84)*v)))/(0.4*((sqrt(0.84)*cosh(0.4*u))^2 + (0.4*sin(sqrt(0.84)*v))^2))
456
        sage: fy = (2*sqrt(0.84)*cosh(0.4*u)*(-(sqrt(0.84)*sin(v)*cos(sqrt(0.84)*v)) + cos(v)*sin(sqrt(0.84)*v)))/(0.4*((sqrt(0.84)*cosh(0.4*u))^2 + (0.4*sin(sqrt(0.84)*v))^2))
457
        sage: fz = -u + (2*0.84*cosh(0.4*u)*sinh(0.4*u))/(0.4*((sqrt(0.84)*cosh(0.4*u))^2 + (0.4*sin(sqrt(0.84)*v))^2))
458
        sage: parametric_plot3d([fx, fy, fz], (u, -13.2, 13.2), (v, -37.4, 37.4), plot_points = [90,90], frame=False, color="green")
459
    
460
    TESTS::
461
    
462
        sage: u, v = var('u,v')
463
        sage: plot3d(u^2-v^2, (u, -1, 1), (u, -1, 1))
464
        Traceback (most recent call last):
465
        ...
466
        ValueError: range variables should be distinct, but there are duplicates
467

468

469
    From Trac #2858::
470
    
471
        sage: parametric_plot3d((u,-u,v), (u,-10,10),(v,-10,10))
472
        sage: f(u)=u; g(v)=v^2; parametric_plot3d((g,f,f), (-10,10),(-10,10))
473

474
    From Trac #5368::
475

476
        sage: x, y = var('x,y')
477
        sage: plot3d(x*y^2 - sin(x), (x,-1,1), (y,-1,1))
478

479
    """
480
    # TODO:
481
    #   * Surface -- behavior of functions not defined everywhere -- see note above
482
    #   * Iterative refinement
483

484
    
485
    # color_function -- (default: "automatic") how to determine the color of curves and surfaces
486
    # color_function_scaling -- (default: True) whether to scale the input to color_function
487
    # exclusions -- (default: "automatic") u points or (u,v) conditions to exclude.
488
    #         (E.g., exclusions could be a function e = lambda u, v: False if u < v else True
489
    # exclusions_style -- (default: None) what to draw at excluded points
490
    # max_recursion -- (default: "automatic") maximum number of recursive subdivisions,
491
    #                   when ...
492
    # mesh -- (default: "automatic") how many mesh divisions in each direction to draw
493
    # mesh_functions -- (default: "automatic") how to determine the placement of mesh divisions
494
    # mesh_shading -- (default: None) how to shade regions between mesh divisions
495
    # plot_range -- (default: "automatic") range of values to include
496

497
    if is_Vector(f):
498
        f = tuple(f)
499

500
    if isinstance(f, (list,tuple)) and len(f) > 0 and isinstance(f[0], (list,tuple)):
501
        return sum([parametric_plot3d(v, urange, vrange, plot_points=plot_points, **kwds) for v in f])
502
            
503
    if not isinstance(f, (tuple, list)) or len(f) != 3:
504
        raise ValueError, "f must be a list, tuple, or vector of length 3"
505

506
    if vrange is None:
507
        if plot_points == "automatic":
508
            plot_points = 75
509
        G = _parametric_plot3d_curve(f, urange, plot_points=plot_points, **kwds)
510
    else:
511
        if plot_points == "automatic":
512
            plot_points = [40,40]
513
        G = _parametric_plot3d_surface(f, urange, vrange, plot_points=plot_points, boundary_style=boundary_style, **kwds)
514
    G._set_extra_kwds(kwds)
515
    return G
516
    
517
def _parametric_plot3d_curve(f, urange, plot_points, **kwds):
518
    r"""
519
    Return a parametric three-dimensional space curve.
520
    This function is used internally by the
521
    :func:`parametric_plot3d` command.
522

523
    There are two ways this function is invoked by
524
    :func:`parametric_plot3d`.
525

526
    - ``parametric_plot3d([f_x, f_y, f_z], (u_min,
527
      u_max))``:
528
      `f_x, f_y, f_z` are three functions and
529
      `u_{\min}` and `u_{\max}` are real numbers
530

531
    - ``parametric_plot3d([f_x, f_y, f_z], (u, u_min,
532
      u_max))``:
533
      `f_x, f_y, f_z` can be viewed as functions of
534
      `u`
535

536
    INPUT:
537

538
    - ``f`` - a 3-tuple of functions or expressions, or vector of size 3
539

540
    - ``urange`` - a 2-tuple (u_min, u_max) or a 3-tuple
541
      (u, u_min, u_max)
542

543
    - ``plot_points`` - (default: "automatic", which is 75) initial
544
      number of sample points in each parameter; an integer.
545

546
    EXAMPLES:
547

548
    We demonstrate each of the two ways of calling this.  See
549
    :func:`parametric_plot3d` for many more examples.
550

551
    We do the first one with a lambda function, which creates a
552
    callable Python function that sends `u` to `u/10`::
553

554
        sage: parametric_plot3d( (sin, cos, lambda u: u/10), (0, 20)) # indirect doctest
555

556
    Now we do the same thing with symbolic expressions::
557

558
        sage: u = var('u')
559
        sage: parametric_plot3d( (sin(u), cos(u), u/10), (u, 0, 20))
560
    """
561
    from sage.plot.misc import setup_for_eval_on_grid
562
    g, ranges = setup_for_eval_on_grid(f, [urange], plot_points)
563
    f_x,f_y,f_z = g
564
    w = [(f_x(u), f_y(u), f_z(u)) for u in xsrange(*ranges[0], include_endpoint=True)]
565
    return line3d(w, **kwds)
566

567
def _parametric_plot3d_surface(f, urange, vrange, plot_points, boundary_style, **kwds):
568
    r"""
569
    Return a parametric three-dimensional space surface.
570
    This function is used internally by the
571
    :func:`parametric_plot3d` command.
572

573
    There are two ways this function is invoked by
574
    :func:`parametric_plot3d`.
575

576
    - ``parametric_plot3d([f_x, f_y, f_z], (u_min, u_max),
577
      (v_min, v_max))``:
578
      `f_x, f_y, f_z` are each functions of two variables
579

580
    - ``parametric_plot3d([f_x, f_y, f_z], (u, u_min,
581
      u_max), (v, v_min, v_max))``:
582
      `f_x, f_y, f_z` can be viewed as functions of
583
      `u` and `v`
584

585
    INPUT:
586

587
    - ``f`` - a 3-tuple of functions or expressions, or vector of size 3
588

589
    - ``urange`` - a 2-tuple (u_min, u_max) or a 3-tuple
590
      (u, u_min, u_max)
591

592
    - ``vrange`` - a 2-tuple (v_min, v_max) or a 3-tuple
593
      (v, v_min, v_max)
594

595
    - ``plot_points`` - (default: "automatic", which is [40,40]
596
      for surfaces) initial number of sample points in each parameter;
597
      a pair of integers.
598

599
    - ``boundary_style`` - (default: None, no boundary) a dict that describes
600
      how to draw the boundaries of regions by giving options that are passed
601
      to the line3d command.
602

603
    EXAMPLES:
604

605
    We demonstrate each of the two ways of calling this.  See
606
    :func:`parametric_plot3d` for many more examples.
607

608
    We do the first one with lambda functions::
609

610
        sage: f = (lambda u,v: cos(u), lambda u,v: sin(u)+cos(v), lambda u,v: sin(v))
611
        sage: parametric_plot3d(f, (0, 2*pi), (-pi, pi)) # indirect doctest
612

613
    Now we do the same thing with symbolic expressions::
614

615
        sage: u, v = var('u,v')
616
        sage: parametric_plot3d((cos(u), sin(u) + cos(v), sin(v)), (u, 0, 2*pi), (v, -pi, pi), mesh=True)
617
    """
618
    from sage.plot.misc import setup_for_eval_on_grid
619
    g, ranges = setup_for_eval_on_grid(f, [urange,vrange], plot_points)
620
    urange = srange(*ranges[0], include_endpoint=True)
621
    vrange = srange(*ranges[1], include_endpoint=True)
622
    G = ParametricSurface(g, (urange, vrange), **kwds)
623
    
624
    if boundary_style is not None:
625
        for u in (urange[0], urange[-1]):
626
            G += line3d([(g[0](u,v), g[1](u,v), g[2](u,v)) for v in vrange], **boundary_style)
627
        for v in (vrange[0], vrange[-1]):
628
            G += line3d([(g[0](u,v), g[1](u,v), g[2](u,v)) for u in urange], **boundary_style)
629
    return G
630

631
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