def 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
.. PLOT::
sphinx_plot(plot3d(lambda x, y: x**2 + y**2, (-2,2), (-2,2)))
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
.. PLOT::
sphinx_plot(plot3d(lambda x, y: x**2 + y**2, (-2,2), (-2,2), adaptive=True))
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
.. PLOT::
sphinx_plot(plot3d(lambda x, y: x**2 + y**2, (-2,2), (-2,2), adaptive=True, initial_depth=5))
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
.. PLOT::
var('x y')
sphinx_plot(plot3d(x**2 + y**2, (x,-2,2), (y,-2,2)))
::
sage: plot3d(sin(x*y), (x, -pi, pi), (y, -pi, pi))
Graphics3d Object
.. PLOT::
var('x y')
sphinx_plot(plot3d(sin(x*y), (x, -pi, pi), (y, -pi, pi)))
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
.. PLOT::
var('x y')
sphinx_plot(plot3d(sin(x**2+y**2),(x,-5,5),(y,-5,5), plot_points=200))
::
sage: plot3d(sin(x^2+y^2),(x,-5,5),(y,-5,5), plot_points=[10,100])
Graphics3d Object
.. PLOT::
var('x y')
sphinx_plot(plot3d(sin(x**2+y**2),(x,-5,5),(y,-5,5), plot_points=[10,100]))
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
.. PLOT::
var('x y')
sphinx_plot(plot3d(sin(x-y)*y*cos(x),(x,-3,3),(y,-3,3), mesh=True))
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
.. PLOT::
x,y=var('x y')
P = plot3d(x+y+sin(x*y), (x,-10,10),(y,-10,10), opacity=0.87, color='blue')
Q = plot3d(x-2*y-cos(x*y),(x,-10,10),(y,-10,10),opacity=0.3,color='red')
sphinx_plot(P+Q)
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
.. PLOT::
L = plot3d(lambda x,y: 0, (-5,5), (-5,5), color="lightblue", opacity=0.8)
P = plot3d(lambda x,y: 4 - x**3 - y**2, (-2,2), (-2,2), color='green')
Q = plot3d(lambda x,y: x**3 + y**2 - 4, (-2,2), (-2,2), color='orange')
sphinx_plot(L+P+Q)
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
.. PLOT::
x, y = var('x y')
sphinx_plot(plot3d(sin(pi*(x**2+y**2))/2,(x,-1,1),(y,-1,1)))
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
.. PLOT::
x, y = var('x y')
sphinx_plot(plot3d( 4*x*exp(-x**2-y**2), (x,-2,2), (y,-2,2)))
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)
.. PLOT::
r, phi, z = var('r phi z')
trans = (r*cos(phi),r*sin(phi),z)
P = plot3d(cos(r),(r,0,17*pi/2),(phi,0,2*pi),transformation=trans,opacity=0.87)
P.aspect_ratio([1,1,2])
sphinx_plot(P)
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
.. PLOT::
r, theta, z = var('r theta z')
cylindrical=(r*cos(theta),r*sin(theta),z)
P = plot3d(z-z+3,(theta,0,pi/2),(z,0,pi/2),transformation=cylindrical)
sphinx_plot(P)
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
"""
if transformation is not None:
params=None
from sage.symbolic.callable import is_CallableSymbolicExpression
# First, determine the parameters for f (from the first item of urange
# and vrange, preferably).
if len(urange) == 3 and len(vrange) == 3:
params = (urange[0], vrange[0])
elif is_CallableSymbolicExpression(f):
params = f.variables()
from sage.modules.vector_callable_symbolic_dense import Vector_callable_symbolic_dense
if isinstance(transformation, (tuple, list,Vector_callable_symbolic_dense)):
if len(transformation)==3:
if params is None:
raise ValueError("must specify independent variable names in the ranges when using generic transformation")
indep_vars = params
elif len(transformation)==4:
indep_vars = transformation[3]
transformation = transformation[0:3]
else:
raise ValueError("unknown transformation type")
# find out which variable is the function variable by
# eliminating the parameter variables.
all_vars = set(sum([list(s.variables()) for s in transformation],[]))
dep_var=all_vars - set(indep_vars)
if len(dep_var)==1:
dep_var = dep_var.pop()
transformation = _ArbitraryCoordinates(transformation, dep_var, indep_vars)
else:
raise ValueError("unable to determine the function variable in the transform")
if isinstance(transformation, _Coordinates):
R = transformation.to_cartesian(f, params)
return parametric_plot3d.parametric_plot3d(R, urange, vrange, **kwds)
else:
raise ValueError('unknown transformation type')
elif adaptive:
P = plot3d_adaptive(f, urange, vrange, **kwds)
else:
u=fast_float_arg(0)
v=fast_float_arg(1)
P=parametric_plot3d.parametric_plot3d((u,v,f), urange, vrange, **kwds)
P.frame_aspect_ratio([1.0,1.0,0.5])
return P
def bezier3d(path, **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
"""
import parametric_plot3d as P3D
from sage.modules.free_module_element import vector
from sage.calculus.calculus import var
p0 = vector(path[0][-1])
t = var('t')
if len(path[0]) > 2:
B = (1-t)**3*vector(path[0][0])+3*t*(1-t)**2*vector(path[0][1])+3*t**2*(1-t)*vector(path[0][-2])+t**3*p0
G = P3D.parametric_plot3d(list(B), (0, 1), color=options['color'], aspect_ratio=options['aspect_ratio'], thickness=options['thickness'], opacity=options['opacity'])
else:
G = line3d([path[0][0], p0], color=options['color'], thickness=options['thickness'], opacity=options['opacity'])
for curve in path[1:]:
if len(curve) > 1:
p1 = vector(curve[0])
p2 = vector(curve[-2])
p3 = vector(curve[-1])
B = (1-t)**3*p0+3*t*(1-t)**2*p1+3*t**2*(1-t)*p2+t**3*p3
G += P3D.parametric_plot3d(list(B), (0, 1), color=options['color'], aspect_ratio=options['aspect_ratio'], thickness=options['thickness'], opacity=options['opacity'])
else:
G += line3d([p0,curve[0]], color=options['color'], thickness=options['thickness'], opacity=options['opacity'])
p0 = curve[-1]
return G
def 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
"""
if transformation is not None:
params = None
from sage.symbolic.callable import is_CallableSymbolicExpression
# First, determine the parameters for f (from the first item of urange
# and vrange, preferably).
if len(urange) == 3 and len(vrange) == 3:
params = (urange[0], vrange[0])
elif is_CallableSymbolicExpression(f):
params = f.variables()
from sage.modules.vector_callable_symbolic_dense import Vector_callable_symbolic_dense
if isinstance(transformation,
(tuple, list, Vector_callable_symbolic_dense)):
if len(transformation) == 3:
if params is None:
raise ValueError(
"must specify independent variable names in the ranges when using generic transformation"
)
indep_vars = params
elif len(transformation) == 4:
indep_vars = transformation[3]
transformation = transformation[0:3]
else:
raise ValueError("unknown transformation type")
# find out which variable is the function variable by
# eliminating the parameter variables.
all_vars = set(
sum([list(s.variables()) for s in transformation], []))
dep_var = all_vars - set(indep_vars)
if len(dep_var) == 1:
dep_var = dep_var.pop()
transformation = _ArbitraryCoordinates(transformation, dep_var,
indep_vars)
else:
raise ValueError(
"unable to determine the function variable in the transform"
)
if isinstance(transformation, _Coordinates):
R = transformation.to_cartesian(f, params)
return parametric_plot3d.parametric_plot3d(R, urange, vrange,
**kwds)
else:
raise ValueError('unknown transformation type')
elif adaptive:
P = plot3d_adaptive(f, urange, vrange, **kwds)
else:
u = fast_float_arg(0)
v = fast_float_arg(1)
P = parametric_plot3d.parametric_plot3d((u, v, f), urange, vrange,
**kwds)
P.frame_aspect_ratio([1.0, 1.0, 0.5])
return P
