def revolution_plot3d(curve,
trange,
phirange=None,
parallel_axis='z',
axis=(0, 0),
print_vector=False,
show_curve=False,
**kwds):
"""
Return a plot of a revolved curve.
There are three ways to call this function:
- ``revolution_plot3d(f,trange)`` where `f` is a function located in the `x z` plane.
- ``revolution_plot3d((f_x,f_z),trange)`` where `(f_x,f_z)` is a parametric curve on the `x z` plane.
- ``revolution_plot3d((f_x,f_y,f_z),trange)`` where `(f_x,f_y,f_z)` can be any parametric curve.
INPUT:
- ``curve`` - A curve to be revolved, specified as a function, a 2-tuple or a 3-tuple.
- ``trange`` - A 3-tuple `(t,t_{\min},t_{\max})` where t is the independent variable of the curve.
- ``phirange`` - A 2-tuple of the form `(\phi_{\min},\phi_{\max})`, (default `(0,\pi)`) that specifies the angle in which the curve is to be revolved.
- ``parallel_axis`` - A string (Either 'x', 'y', or 'z') that specifies the coordinate axis parallel to the revolution axis.
- ``axis`` - A 2-tuple that specifies the position of the revolution axis. If parallel is:
- 'z' - then axis is the point in which the revolution axis intersects the `x y` plane.
- 'x' - then axis is the point in which the revolution axis intersects the `y z` plane.
- 'y' - then axis is the point in which the revolution axis intersects the `x z` plane.
- ``print_vector`` - If True, the parametrization of the surface of revolution will be printed.
- ``show_curve`` - If True, the curve will be displayed.
EXAMPLES:
Let's revolve a simple function around different axes::
sage: u = var('u')
sage: f=u^2
sage: revolution_plot3d(f,(u,0,2),show_curve=True,opacity=0.7).show(aspect_ratio=(1,1,1))
If we move slightly the axis, we get a goblet-like surface::
sage: revolution_plot3d(f,(u,0,2),axis=(1,0.2),show_curve=True,opacity=0.5).show(aspect_ratio=(1,1,1))
A common problem in calculus books, find the volume within the following revolution solid::
sage: line=u
sage: parabola=u^2
sage: sur1=revolution_plot3d(line,(u,0,1),opacity=0.5,rgbcolor=(1,0.5,0),show_curve=True,parallel_axis='x')
sage: sur2=revolution_plot3d(parabola,(u,0,1),opacity=0.5,rgbcolor=(0,1,0),show_curve=True,parallel_axis='x')
sage: (sur1+sur2).show()
Now let's revolve a parametrically defined circle. We can play with the topology of the surface by changing the axis, an axis in `(0,0)` (as the previous one) will produce a sphere-like surface::
sage: u = var('u')
sage: circle=(cos(u),sin(u))
sage: revolution_plot3d(circle,(u,0,2*pi),axis=(0,0),show_curve=True,opacity=0.5).show(aspect_ratio=(1,1,1))
An axis on `(0,y)` will produce a cylinder-like surface::
sage: revolution_plot3d(circle,(u,0,2*pi),axis=(0,2),show_curve=True,opacity=0.5).show(aspect_ratio=(1,1,1))
And any other axis will produce a torus-like surface::
sage: revolution_plot3d(circle,(u,0,2*pi),axis=(2,0),show_curve=True,opacity=0.5).show(aspect_ratio=(1,1,1))
Now, we can get another goblet-like surface by revolving a curve in 3d::
sage: u = var('u')
sage: curve=(u,cos(4*u),u^2)
sage: revolution_plot3d(curve,(u,0,2),show_curve=True,parallel_axis='z',axis=(1,.2),opacity=0.5).show(aspect_ratio=(1,1,1))
A curvy curve with only a quarter turn::
sage: u = var('u')
sage: curve=(sin(3*u),.8*cos(4*u),cos(u))
sage: revolution_plot3d(curve,(u,0,pi),(0,pi/2),show_curve=True,parallel_axis='z',opacity=0.5).show(aspect_ratio=(1,1,1),frame=False)
"""
from sage.symbolic.ring import SR
from sage.symbolic.constants import pi
from sage.functions.other import sqrt
from sage.functions.trig import sin
from sage.functions.trig import cos
from sage.functions.trig import atan2
if parallel_axis not in ['x', 'y', 'z']:
raise ValueError("parallel_axis must be either 'x', 'y', or 'z'.")
vart = trange[0]
if str(vart) == 'phi':
phi = SR.var('fi')
else:
phi = SR.var('phi')
if phirange is None: #this if-else provides a phirange
phirange = (phi, 0, 2 * pi)
elif len(phirange) == 3:
phi = phirange[0]
pass
else:
phirange = (phi, phirange[0], phirange[1])
if isinstance(curve, tuple) or isinstance(curve, list):
#this if-else provides a vector v to be plotted
#if curve is a tuple or a list of length 2, it is interpreted as a parametric curve
#in the x-z plane.
#if it is of length 3 it is interpreted as a parametric curve in 3d space
if len(curve) == 2:
x = curve[0]
y = 0
z = curve[1]
elif len(curve) == 3:
x = curve[0]
y = curve[1]
z = curve[2]
else:
x = vart
y = 0
z = curve
if parallel_axis == 'z':
x0 = axis[0]
y0 = axis[1]
# (0,0) must be handled separately for the phase value
phase = 0
if x0 != 0 or y0 != 0:
phase = atan2((y - y0), (x - x0))
R = sqrt((x - x0)**2 + (y - y0)**2)
v = (R * cos(phi + phase) + x0, R * sin(phi + phase) + y0, z)
elif parallel_axis == 'x':
y0 = axis[0]
z0 = axis[1]
# (0,0) must be handled separately for the phase value
phase = 0
if z0 != 0 or y0 != 0:
phase = atan2((z - z0), (y - y0))
R = sqrt((y - y0)**2 + (z - z0)**2)
v = (x, R * cos(phi + phase) + y0, R * sin(phi + phase) + z0)
elif parallel_axis == 'y':
x0 = axis[0]
z0 = axis[1]
# (0,0) must be handled separately for the phase value
phase = 0
if z0 != 0 or x0 != 0:
phase = atan2((z - z0), (x - x0))
R = sqrt((x - x0)**2 + (z - z0)**2)
v = (R * cos(phi + phase) + x0, y, R * sin(phi + phase) + z0)
if print_vector:
print(v)
if show_curve:
curveplot = parametric_plot3d((x, y, z),
trange,
thickness=2,
rgbcolor=(1, 0, 0))
return parametric_plot3d(v, trange, phirange, **kwds) + curveplot
return parametric_plot3d(v, trange, phirange, **kwds)
def revolution_plot3d(curve,
trange,
phirange=None,
parallel_axis='z',
axis=(0, 0),
print_vector=False,
show_curve=False,
**kwds):
r"""
Return a plot of a revolved curve.
There are three ways to call this function:
- ``revolution_plot3d(f,trange)`` where `f` is a function located in the `x z` plane.
- ``revolution_plot3d((f_x,f_z),trange)`` where `(f_x,f_z)` is a parametric curve on the `x z` plane.
- ``revolution_plot3d((f_x,f_y,f_z),trange)`` where `(f_x,f_y,f_z)` can be any parametric curve.
INPUT:
- ``curve`` - A curve to be revolved, specified as a function, a 2-tuple or a 3-tuple.
- ``trange`` - A 3-tuple `(t,t_{\min},t_{\max})` where t is the independent variable of the curve.
- ``phirange`` - A 2-tuple of the form `(\phi_{\min},\phi_{\max})`, (default `(0,\pi)`) that specifies the angle in which the curve is to be revolved.
- ``parallel_axis`` - A string (Either 'x', 'y', or 'z') that specifies the coordinate axis parallel to the revolution axis.
- ``axis`` - A 2-tuple that specifies the position of the revolution axis. If parallel is:
- 'z' - then axis is the point in which the revolution axis intersects the `x y` plane.
- 'x' - then axis is the point in which the revolution axis intersects the `y z` plane.
- 'y' - then axis is the point in which the revolution axis intersects the `x z` plane.
- ``print_vector`` - If True, the parametrization of the surface of revolution will be printed.
- ``show_curve`` - If True, the curve will be displayed.
EXAMPLES:
Let's revolve a simple function around different axes::
sage: u = var('u')
sage: f = u^2
sage: revolution_plot3d(f, (u,0,2), show_curve=True, opacity=0.7).show(aspect_ratio=(1,1,1))
.. PLOT::
u = var('u')
f = u**2
P = revolution_plot3d(f, (u,0,2), show_curve=True, opacity=0.7).plot()
sphinx_plot(P)
If we move slightly the axis, we get a goblet-like surface::
sage: revolution_plot3d(f, (u,0,2), axis=(1,0.2), show_curve=True, opacity=0.5).show(aspect_ratio=(1,1,1))
.. PLOT::
u = var('u')
f = u**2
P = revolution_plot3d(f, (u,0,2), axis=(1,0.2), show_curve=True, opacity=0.5).plot()
sphinx_plot(P)
A common problem in calculus books, find the volume within the following revolution solid::
sage: line = u
sage: parabola = u^2
sage: sur1 = revolution_plot3d(line, (u,0,1), opacity=0.5, rgbcolor=(1,0.5,0), show_curve=True, parallel_axis='x')
sage: sur2 = revolution_plot3d(parabola, (u,0,1), opacity=0.5, rgbcolor=(0,1,0), show_curve=True, parallel_axis='x')
sage: (sur1+sur2).show()
.. PLOT::
u = var('u')
line = u
parabola = u**2
sur1 = revolution_plot3d(line, (u,0,1), opacity=0.5, rgbcolor=(1,0.5,0), show_curve=True, parallel_axis='x')
sur2 = revolution_plot3d(parabola, (u,0,1), opacity=0.5, rgbcolor=(0,1,0), show_curve=True, parallel_axis='x')
P = sur1 + sur2
sphinx_plot(P)
Now let's revolve a parametrically defined circle. We can play with the topology of the surface by changing the axis,
an axis in `(0,0)` (as the previous one) will produce a sphere-like surface::
sage: u = var('u')
sage: circle = (cos(u), sin(u))
sage: revolution_plot3d(circle, (u,0,2*pi), axis=(0,0), show_curve=True, opacity=0.5).show(aspect_ratio=(1,1,1))
.. PLOT::
u = var('u')
circle = (cos(u), sin(u))
P = revolution_plot3d(circle, (u,0,2*pi), axis=(0,0), show_curve=True, opacity=0.5)
sphinx_plot(P)
An axis on `(0,y)` will produce a cylinder-like surface::
sage: revolution_plot3d(circle, (u,0,2*pi), axis=(0,2), show_curve=True, opacity=0.5).show(aspect_ratio=(1,1,1))
.. PLOT::
u = var('u')
circle = (cos(u), sin(u))
P = revolution_plot3d(circle, (u,0,2*pi), axis=(0,2), show_curve=True, opacity=0.5)
sphinx_plot(P)
And any other axis will produce a torus-like surface::
sage: revolution_plot3d(circle, (u,0,2*pi), axis=(2,0), show_curve=True, opacity=0.5).show(aspect_ratio=(1,1,1))
.. PLOT::
u = var('u')
circle = (cos(u), sin(u))
P = revolution_plot3d(circle, (u,0,2*pi), axis=(2,0), show_curve=True, opacity=0.5)
sphinx_plot(P)
Now, we can get another goblet-like surface by revolving a curve in 3d::
sage: u = var('u')
sage: curve = (u, cos(4*u), u^2)
sage: P = revolution_plot3d(curve, (u,0,2), show_curve=True, parallel_axis='z',axis=(1,.2), opacity=0.5)
sage: P.show(aspect_ratio=(1,1,1))
.. PLOT::
u = var('u')
curve = (u, cos(4*u), u**2)
P = revolution_plot3d(curve, (u,0,2), show_curve=True, parallel_axis='z', axis=(1,.2), opacity=0.5)
sphinx_plot(P)
A curvy curve with only a quarter turn::
sage: u = var('u')
sage: curve = (sin(3*u), .8*cos(4*u), cos(u))
sage: revolution_plot3d(curve, (u,0,pi), (0,pi/2), show_curve=True, parallel_axis='z', opacity=0.5).show(aspect_ratio=(1,1,1),frame=False)
.. PLOT::
u = var('u')
curve = (sin(3*u), .8*cos(4*u), cos(u))
P = revolution_plot3d(curve, (u,0,pi), (0,pi/2), show_curve=True, parallel_axis='z', opacity=0.5)
sphinx_plot(P)
One can also color the surface using a coloring function of two
parameters and a colormap as follows. Note that the coloring
function must take values in the interval [0,1]. ::
sage: u, phi = var('u,phi')
sage: def cf(u,phi): return sin(phi+u) ^ 2
sage: curve = (1+u^2/4, 0, u)
sage: revolution_plot3d(curve, (u,-2,2), (0,2*pi), parallel_axis='z', color=(cf, colormaps.PiYG)).show(aspect_ratio=(1,1,1))
.. PLOT::
u, phi = var('u,phi')
def cf(u, phi): return sin(phi+u) ** 2
curve = (1+u**2/4, 0, u)
P = revolution_plot3d(curve, (u,-2,2), (0,2*pi), parallel_axis='z', color=(cf, colormaps.PiYG))
sphinx_plot(P)
The first parameter of the coloring function will be identified with the
parameter of the curve, and the second with the angle parameter.
.. WARNING::
This kind of coloring using a colormap can be visualized using
Jmol, Tachyon (option ``viewer='tachyon'``) and Canvas3D
(option ``viewer='canvas3d'`` in the notebook).
Another colored example, illustrating that one can use (colormap, color function) instead of (color function, colormap)::
sage: u, phi = var('u,phi')
sage: def cf(u, phi): return float(2 * u / pi) % 1
sage: curve = (sin(u), 0, u)
sage: revolution_plot3d(curve, (u,0,pi), (0,2*pi), parallel_axis
....: ='z', color=(colormaps.brg, cf)).show(aspect_ratio=1)
.. PLOT::
u, phi = var('u,phi')
def cf(u, phi): return float(2 * u / pi) % 1
curve = (sin(u), 0, u)
P = revolution_plot3d(curve, (u,0,pi), (0,2*pi), parallel_axis='z', color=(colormaps.brg, cf))
sphinx_plot(P)
"""
from sage.symbolic.ring import SR
from sage.symbolic.constants import pi
from sage.misc.functional import sqrt
from sage.functions.trig import sin
from sage.functions.trig import cos
from sage.functions.trig import atan2
if parallel_axis not in ['x', 'y', 'z']:
raise ValueError("parallel_axis must be either 'x', 'y', or 'z'.")
vart = trange[0]
if str(vart) == 'phi':
phi = SR.var('fi')
else:
phi = SR.var('phi')
if phirange is None: # this if-else provides a phirange
phirange = (phi, 0, 2 * pi)
elif len(phirange) == 3:
phi = phirange[0]
else:
phirange = (phi, phirange[0], phirange[1])
if isinstance(curve, (tuple, list)):
#this if-else provides a vector v to be plotted
#if curve is a tuple or a list of length 2, it is interpreted as a parametric curve
#in the x-z plane.
#if it is of length 3 it is interpreted as a parametric curve in 3d space
if len(curve) == 2:
x = curve[0]
y = 0
z = curve[1]
elif len(curve) == 3:
x = curve[0]
y = curve[1]
z = curve[2]
else:
x = vart
y = 0
z = curve
phase = 0
if parallel_axis == 'z':
x0 = axis[0]
y0 = axis[1]
# (0,0) must be handled separately for the phase value
if x0 != 0 or y0 != 0:
phase = atan2(y - y0, x - x0)
R = sqrt((x - x0)**2 + (y - y0)**2)
v = (R * cos(phi + phase) + x0, R * sin(phi + phase) + y0, z)
elif parallel_axis == 'x':
y0 = axis[0]
z0 = axis[1]
# (0,0) must be handled separately for the phase value
if z0 != 0 or y0 != 0:
phase = atan2(z - z0, y - y0)
R = sqrt((y - y0)**2 + (z - z0)**2)
v = (x, R * cos(phi + phase) + y0, R * sin(phi + phase) + z0)
elif parallel_axis == 'y':
x0 = axis[0]
z0 = axis[1]
# (0,0) must be handled separately for the phase value
if z0 != 0 or x0 != 0:
phase = atan2(z - z0, x - x0)
R = sqrt((x - x0)**2 + (z - z0)**2)
v = (R * cos(phi + phase) + x0, y, R * sin(phi + phase) + z0)
if print_vector:
print(v)
if show_curve:
curveplot = parametric_plot3d((x, y, z),
trange,
thickness=2,
rgbcolor=(1, 0, 0))
return parametric_plot3d(v, trange, phirange, **kwds) + curveplot
return parametric_plot3d(v, trange, phirange, **kwds)
def _transition_spher_stereo(self):
r"""
Initialize the transition map between spherical and stereographic
coordinates.
TESTS::
sage: S1 = manifolds.Sphere(1)
sage: spher = S1.spherical_coordinates(); spher
Chart (A, (phi,))
sage: A = spher.domain()
sage: stereoN = S1.stereographic_coordinates(pole='north'); stereoN
Chart (S^1-{NP}, (y1,))
sage: S1.coord_change(spher, stereoN.restrict(A))
Change of coordinates from Chart (A, (phi,)) to Chart (A, (y1,))
"""
# speed-up via simplification method...
self.set_simplify_function(lambda expr: expr.simplify())
# configure preexisting charts...
W = self._stereoN_dom.intersection(self._stereoS_dom)
A = self._spher_dom
stereoN, stereoS = self._coordinates['stereographic'][:]
coordN = stereoN[:]
coordS = stereoS[:]
rstN = (coordN[0] != 0, )
rstS = (coordS[0] != 0, )
if self._dim > 1:
rstN += (coordN[0] > 0, )
rstS += (coordS[0] > 0, )
stereoN_A = stereoN.restrict(A, rstN)
stereoS_A = stereoS.restrict(A, rstS)
self._coord_changes[(stereoN.restrict(W),
stereoS.restrict(W))].restrict(A)
self._coord_changes[(stereoS.restrict(W),
stereoN.restrict(W))].restrict(A)
spher = self._coordinates['spherical'][0]
R = self._radius
n = self._dim
# transition: spher to stereoN...
imm = self.embedding()
cart = self._ambient.cartesian_coordinates()
# get ambient coordinates and shift to coordinate origin:
x = self._shift_coords(imm.expr(spher, cart), s='-')
coordfunc = [(R * x[i]) / (R - x[-1]) for i in range(n)]
# define transition map:
spher_to_stereoN = spher.transition_map(stereoN_A, coordfunc)
# transition: stereoN to spher...
from sage.functions.trig import acos, atan2
from sage.misc.functional import sqrt
# get ambient coordinates and shift to coordinate origin:
x = self._shift_coords(imm.expr(stereoN, cart), s='-')
coordfunc = [atan2(x[1], x[0])]
for k in range(2, n + 1):
c = acos(x[k] / sqrt(sum(x[i]**2 for i in range(k + 1))))
coordfunc.append(c)
coordfunc = reversed(coordfunc)
spher_to_stereoN.set_inverse(*coordfunc, check=False)
# transition spher <-> stereoS...
stereoN_to_S_A = self.coord_change(stereoN_A, stereoS_A)
stereoN_to_S_A * spher_to_stereoN # generates spher_to_stereoS
stereoS_to_N_A = self.coord_change(stereoS_A, stereoN_A)
spher_to_stereoN.inverse(
) * stereoS_to_N_A # generates stereoS_to_spher
def revolution_plot3d(curve,trange,phirange=None,parallel_axis='z',axis=(0,0),print_vector=False,show_curve=False,**kwds):
"""
Return a plot of a revolved curve.
There are three ways to call this function:
- ``revolution_plot3d(f,trange)`` where `f` is a function located in the `x z` plane.
- ``revolution_plot3d((f_x,f_z),trange)`` where `(f_x,f_z)` is a parametric curve on the `x z` plane.
- ``revolution_plot3d((f_x,f_y,f_z),trange)`` where `(f_x,f_y,f_z)` can be any parametric curve.
INPUT:
- ``curve`` - A curve to be revolved, specified as a function, a 2-tuple or a 3-tuple.
- ``trange`` - A 3-tuple `(t,t_{\min},t_{\max})` where t is the independent variable of the curve.
- ``phirange`` - A 2-tuple of the form `(\phi_{\min},\phi_{\max})`, (default `(0,\pi)`) that specifies the angle in which the curve is to be revolved.
- ``parallel_axis`` - A string (Either 'x', 'y', or 'z') that specifies the coordinate axis parallel to the revolution axis.
- ``axis`` - A 2-tuple that specifies the position of the revolution axis. If parallel is:
- 'z' - then axis is the point in which the revolution axis intersects the `x y` plane.
- 'x' - then axis is the point in which the revolution axis intersects the `y z` plane.
- 'y' - then axis is the point in which the revolution axis intersects the `x z` plane.
- ``print_vector`` - If True, the parametrization of the surface of revolution will be printed.
- ``show_curve`` - If True, the curve will be displayed.
EXAMPLES:
Let's revolve a simple function around different axes::
sage: u = var('u')
sage: f=u^2
sage: revolution_plot3d(f,(u,0,2),show_curve=True,opacity=0.7).show(aspect_ratio=(1,1,1))
If we move slightly the axis, we get a goblet-like surface::
sage: revolution_plot3d(f,(u,0,2),axis=(1,0.2),show_curve=True,opacity=0.5).show(aspect_ratio=(1,1,1))
A common problem in calculus books, find the volume within the following revolution solid::
sage: line=u
sage: parabola=u^2
sage: sur1=revolution_plot3d(line,(u,0,1),opacity=0.5,rgbcolor=(1,0.5,0),show_curve=True,parallel_axis='x')
sage: sur2=revolution_plot3d(parabola,(u,0,1),opacity=0.5,rgbcolor=(0,1,0),show_curve=True,parallel_axis='x')
sage: (sur1+sur2).show()
Now let's revolve a parametrically defined circle. We can play with the topology of the surface by changing the axis, an axis in `(0,0)` (as the previous one) will produce a sphere-like surface::
sage: u = var('u')
sage: circle=(cos(u),sin(u))
sage: revolution_plot3d(circle,(u,0,2*pi),axis=(0,0),show_curve=True,opacity=0.5).show(aspect_ratio=(1,1,1))
An axis on `(0,y)` will produce a cylinder-like surface::
sage: revolution_plot3d(circle,(u,0,2*pi),axis=(0,2),show_curve=True,opacity=0.5).show(aspect_ratio=(1,1,1))
And any other axis will produce a torus-like surface::
sage: revolution_plot3d(circle,(u,0,2*pi),axis=(2,0),show_curve=True,opacity=0.5).show(aspect_ratio=(1,1,1))
Now, we can get another goblet-like surface by revolving a curve in 3d::
sage: u = var('u')
sage: curve=(u,cos(4*u),u^2)
sage: revolution_plot3d(curve,(u,0,2),show_curve=True,parallel_axis='z',axis=(1,.2),opacity=0.5).show(aspect_ratio=(1,1,1))
A curvy curve with only a quarter turn::
sage: u = var('u')
sage: curve=(sin(3*u),.8*cos(4*u),cos(u))
sage: revolution_plot3d(curve,(u,0,pi),(0,pi/2),show_curve=True,parallel_axis='z',opacity=0.5).show(aspect_ratio=(1,1,1),frame=False)
"""
from sage.symbolic.ring import var
from sage.symbolic.constants import pi
from sage.functions.other import sqrt
from sage.functions.trig import sin
from sage.functions.trig import cos
from sage.functions.trig import atan2
if parallel_axis not in ['x','y','z']:
raise ValueError, "parallel_axis must be either 'x', 'y', or 'z'."
vart=trange[0]
if str(vart)=='phi':
phi=var('fi')
else:
phi=var('phi')
if phirange is None:#this if-else provides a phirange
phirange=(phi,0,2*pi)
elif len(phirange)==3:
phi=phirange[0]
pass
else:
phirange=(phi,phirange[0],phirange[1])
if isinstance(curve,tuple) or isinstance(curve,list):
#this if-else provides a vector v to be plotted
#if curve is a tuple or a list of length 2, it is interpreted as a parametric curve
#in the x-z plane.
#if it is of length 3 it is interpreted as a parametric curve in 3d space
if len(curve) ==2:
x=curve[0]
y=0
z=curve[1]
elif len(curve)==3:
x=curve[0]
y=curve[1]
z=curve[2]
else:
x=vart
y=0
z=curve
if parallel_axis=='z':
x0=axis[0]
y0=axis[1]
# (0,0) must be handled separately for the phase value
phase=0
if x0!=0 or y0!=0:
phase=atan2((y-y0),(x-x0))
R=sqrt((x-x0)**2+(y-y0)**2)
v=(R*cos(phi+phase)+x0,R*sin(phi+phase)+y0,z)
elif parallel_axis=='x':
y0=axis[0]
z0=axis[1]
# (0,0) must be handled separately for the phase value
phase=0
if z0!=0 or y0!=0:
phase=atan2((z-z0),(y-y0))
R=sqrt((y-y0)**2+(z-z0)**2)
v=(x,R*cos(phi+phase)+y0,R*sin(phi+phase)+z0)
elif parallel_axis=='y':
x0=axis[0]
z0=axis[1]
# (0,0) must be handled separately for the phase value
phase=0
if z0!=0 or x0!=0:
phase=atan2((z-z0),(x-x0))
R=sqrt((x-x0)**2+(z-z0)**2)
v=(R*cos(phi+phase)+x0,y,R*sin(phi+phase)+z0)
if print_vector:
print v
if show_curve:
curveplot=parametric_plot3d((x,y,z),trange,thickness=2,rgbcolor=(1,0,0))
return parametric_plot3d(v,trange,phirange,**kwds)+curveplot
return parametric_plot3d(v,trange,phirange,**kwds)
def Sphere(dim=None, radius=1, names=None, stereo2d=False, stereo_lim=None):
"""
Generate a sphere embedded in Euclidean space.
The shortcut operator ``.<,>`` can be used to specify the coordinates.
INPUT:
- ``dim`` -- (optional) the dimension of the sphere; if not specified,
equals to the number of coordinate names
- ``radius`` -- (default: ``1``) radius of the sphere
- ``names`` -- (default: ``None``) name of the coordinates,
automatically set by the shortcut operator
- ``stereo2d`` -- (default: ``False``) if ``True``, defines only the
stereographic charts, only implemented in 2d
- ``stereo_lim`` -- (default: ``None``) parameter used to restrict the
span of the stereographic charts, so that they don't cover the whole
sphere; valid domain will be ``x**2 + y**2 < stereo_lim**2``
OUTPUT:
- Riemannian manifold
EXAMPLES::
sage: S.<th, ph> = manifolds.Sphere()
sage: S
2-dimensional Riemannian submanifold S embedded in the Euclidean
space E^3
sage: S.atlas()
[Chart (S, (th, ph))]
sage: S.metric().display()
gamma = dth*dth + sin(th)^2 dph*dph
sage: S = manifolds.Sphere(2, stereo2d=True) # long time
sage: S # long time
2-dimensional Riemannian submanifold S embedded in the Euclidean
space E^3
sage: S.metric().display() # long time
gamma = 4/(x^4 + y^4 + 2*(x^2 + 1)*y^2 + 2*x^2 + 1) dx*dx
+ 4/(x^4 + y^4 + 2*(x^2 + 1)*y^2 + 2*x^2 + 1) dy*dy
"""
from sage.functions.trig import cos, sin, atan, atan2
from sage.functions.other import sqrt
from sage.symbolic.constants import pi
from sage.misc.misc_c import prod
from sage.manifolds.manifold import Manifold
from sage.manifolds.differentiable.euclidean import EuclideanSpace
if dim is None:
if names is None:
raise ValueError("either the names or the dimension must be specified")
dim = len(names)
else:
if names is not None and dim != len(names):
raise ValueError("the number of coordinates does not match the dimension")
if stereo2d:
if dim != 2:
raise NotImplementedError("stereographic charts only "
"implemented for 2d spheres")
E = EuclideanSpace(3, names=("X", "Y", "Z"))
S2 = Manifold(dim, 'S', ambient=E, structure='Riemannian')
U = S2.open_subset('U')
V = S2.open_subset('V')
stereoN = U.chart(names=("x", "y"))
x, y = stereoN[:]
stereoS = V.chart(names=("xp", "yp"))
xp, yp = stereoS[:]
if stereo_lim is not None:
stereoN.add_restrictions(x**2+y**2 < stereo_lim**2)
stereoS.add_restrictions(xp**2+yp**2 < stereo_lim**2)
stereoN_to_S = stereoN.transition_map(stereoS,
(x / (x**2 + y**2), y / (x**2 + y**2)), intersection_name='W',
restrictions1=x**2 + y**2 != 0, restrictions2=xp**2+yp**2!=0)
stereoN_to_S.set_inverse(xp / (xp**2 + yp**2), yp / (xp**2 + yp**2),
check=False)
W = U.intersection(V)
stereoN_W = stereoN.restrict(W)
stereoS_W = stereoS.restrict(W)
A = W.open_subset('A', coord_def={stereoN_W: (y != 0, x < 0),
stereoS_W: (yp != 0, xp < 0)})
stereoN_A = stereoN_W.restrict(A)
if names is None:
names = tuple(["phi_{}:(0,pi)".format(i) for i in range(dim - 1)] +
["phi_{}:(-pi,pi):periodic".format(dim - 1)])
else:
names = tuple([names[i] + ":(0,pi)" for i in range(dim - 1)] +
[names[dim - 1] + ":(-pi,pi):periodic"])
spher = A.chart(names=names)
th, ph = spher[:]
spher_to_stereoN = spher.transition_map(stereoN_A, (sin(th)*cos(ph) / (1-cos(th)),
sin(th)*sin(ph) / (1-cos(th))))
spher_to_stereoN.set_inverse(2*atan(1/sqrt(x**2+y**2)), atan2(-y, -x)+pi,
check=False)
stereoN_to_S_A = stereoN_to_S.restrict(A)
stereoN_to_S_A * spher_to_stereoN # generates spher_to_stereoS
stereoS_to_N_A = stereoN_to_S.inverse().restrict(A)
spher_to_stereoN.inverse() * stereoS_to_N_A # generates stereoS_to_spher
coordfunc1 = [sin(th)*cos(ph), sin(th)*sin(ph), cos(th)]
coordfunc2 = [2*x/(1+x**2+y**2), 2*y/(1+x**2+y**2), (x**2+y**2-1)/(1+x**2+y**2)]
coordfunc3 = [2*xp/(1+xp**2+yp**2), 2*yp/(1+xp**2+yp**2),(1-xp**2-yp**2)/(1+xp**2+yp**2)]
imm = S2.diff_map(E, {(spher, E.default_chart()): coordfunc1,
(stereoN, E.default_chart()): coordfunc2,
(stereoS, E.default_chart()): coordfunc3})
S2.set_embedding(imm)
S2.induced_metric()
return S2
if dim != 2:
raise NotImplementedError("only implemented for 2 dimensional spheres")
E = EuclideanSpace(3, symbols='X Y Z')
M = Manifold(dim, 'S', ambient=E, structure='Riemannian')
if names is None:
names = tuple(["phi_{}:(0,pi)".format(i) for i in range(dim-1)] +
["phi_{}:(-pi,pi):periodic".format(dim-1)])
else:
names = tuple([names[i]+":(0,pi)"for i in range(dim - 1)] +
[names[dim-1]+":(-pi,pi):periodic"])
C = M.chart(names=names)
M._first_ngens = C._first_ngens
phi = M._first_ngens(dim)[:]
coordfunc = ([radius * prod(sin(phi[j]) for j in range(dim))] +
[radius * cos(phi[i]) * prod(sin(phi[j]) for j in range(i))
for i in range(dim)])
imm = M.diff_map(E, coordfunc)
M.set_embedding(imm)
M.induced_metric()
return M
