def unify_arguments(funcs):
"""
Return a tuple of variables of the functions, as well as the
number of "free" variables (i.e., variables that defined in a
callable function).
INPUT:
- ``funcs`` -- a list of functions; these can be symbolic
expressions, polynomials, etc
OUTPUT: functions, expected arguments
- A tuple of variables in the functions
- A tuple of variables that were "free" in the functions
EXAMPLES::
sage: x,y,z=var('x,y,z')
sage: f(x,y)=x+y-z
sage: g(x,y)=x+y
sage: h(y)=-y
sage: sage.plot.misc.unify_arguments((f,g,h))
((x, y, z), (z,))
sage: sage.plot.misc.unify_arguments((g,h))
((x, y), ())
sage: sage.plot.misc.unify_arguments((f,z))
((x, y, z), (z,))
sage: sage.plot.misc.unify_arguments((h,z))
((y, z), (z,))
sage: sage.plot.misc.unify_arguments((x+y,x-y))
((x, y), (x, y))
"""
from sage.symbolic.callable import is_CallableSymbolicExpression
vars = set()
free_variables = set()
if not isinstance(funcs, (list, tuple)):
funcs = [funcs]
for f in funcs:
if is_CallableSymbolicExpression(f):
f_args = set(f.arguments())
vars.update(f_args)
else:
f_args = set()
try:
free_vars = set(f.variables()).difference(f_args)
vars.update(free_vars)
free_variables.update(free_vars)
except AttributeError:
# we probably have a constant
pass
return tuple(sorted(vars, key=lambda x: str(x))), tuple(
sorted(free_variables, key=lambda x: str(x)))
def unify_arguments(funcs):
"""
Returns a tuple of variables of the functions, as well as the
number of "free" variables (i.e., variables that defined in a
callable function).
INPUT:
- ``funcs`` -- a list of functions; these can be symbolic
expressions, polynomials, etc
OUTPUT: functions, expected arguments
- A tuple of variables in the functions
- A tuple of variables that were "free" in the functions
EXAMPLES:
sage: x,y,z=var('x,y,z')
sage: f(x,y)=x+y-z
sage: g(x,y)=x+y
sage: h(y)=-y
sage: sage.plot.misc.unify_arguments((f,g,h))
((x, y, z), (z,))
sage: sage.plot.misc.unify_arguments((g,h))
((x, y), ())
sage: sage.plot.misc.unify_arguments((f,z))
((x, y, z), (z,))
sage: sage.plot.misc.unify_arguments((h,z))
((y, z), (z,))
sage: sage.plot.misc.unify_arguments((x+y,x-y))
((x, y), (x, y))
"""
from sage.symbolic.callable import is_CallableSymbolicExpression
vars=set()
free_variables=set()
if not isinstance(funcs, (list, tuple)):
funcs=[funcs]
for f in funcs:
if is_CallableSymbolicExpression(f):
f_args=set(f.arguments())
vars.update(f_args)
else:
f_args=set()
try:
free_vars = set(f.variables()).difference(f_args)
vars.update(free_vars)
free_variables.update(free_vars)
except AttributeError:
# we probably have a constant
pass
return tuple(sorted(vars, key=lambda x: str(x))), tuple(sorted(free_variables, key=lambda x: str(x)))
def adapt_to_callable(f, nargs=None):
"""
Tries to make every function in f into a (fast) callable
function, returning a new list of functions and the expected
arguments.
INPUT:
- ``f`` -- a list of functions; these can be symbolic expressions,
polynomials, etc
- ``nargs`` -- number of input args to have in the output functions
OUTPUT: functions, expected arguments
"""
from sage.misc.misc import deprecation
deprecation(
"adapt_to_callable is a deprecated function. Please use functions from sage.misc.plot instead."
)
try:
from sage.symbolic.callable import is_CallableSymbolicExpression
if sum([is_CallableSymbolicExpression(z) for z in f]):
# Sum to get common universe; this works since f is
# callable, and summing gets the arguments in the right
# order.
vars = sum(f).args()
else:
# Otherwise any free variable names in any order
try:
vars = tuple(sorted(set(sum([z.variables() for z in f], ()))))
if len(vars) > 1:
from sage.misc.misc import deprecation
deprecation(
"Substitution using function-call syntax and unnamed arguments is deprecated and will be removed from a future release of Sage; you can use named arguments instead, like EXPR(x=..., y=...)"
)
except AttributeError:
vars = ()
f = [fast_float_constant(x) for x in f]
except TypeError:
vars = ()
f = [fast_float_constant(x) for x in f]
if nargs is not None and len(vars) != nargs:
vars = (vars + ('_', ) * nargs)[:nargs]
return fast_float(f, *vars), vars
def adapt_to_callable(f, nargs=None):
"""
Tries to make every function in f into a (fast) callable
function, returning a new list of functions and the expected
arguments.
INPUT:
- ``f`` -- a list of functions; these can be symbolic expressions,
polynomials, etc
- ``nargs`` -- number of input args to have in the output functions
OUTPUT: functions, expected arguments
"""
from sage.misc.misc import deprecation
deprecation("adapt_to_callable is a deprecated function. Please use functions from sage.misc.plot instead.")
try:
from sage.symbolic.callable import is_CallableSymbolicExpression
if sum([is_CallableSymbolicExpression(z) for z in f]):
# Sum to get common universe; this works since f is
# callable, and summing gets the arguments in the right
# order.
vars = sum(f).args()
else:
# Otherwise any free variable names in any order
try:
vars = tuple(sorted(set(sum( [z.variables() for z in f], ()) )))
if len(vars) > 1:
from sage.misc.misc import deprecation
deprecation("Substitution using function-call syntax and unnamed arguments is deprecated and will be removed from a future release of Sage; you can use named arguments instead, like EXPR(x=..., y=...)")
except AttributeError:
vars = ()
f = [fast_float_constant(x) for x in f]
except TypeError:
vars = ()
f = [fast_float_constant(x) for x in f]
if nargs is not None and len(vars) != nargs:
vars = (vars + ('_',)*nargs)[:nargs]
return fast_float(f, *vars), vars
def 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
.. 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 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