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 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 plot_hyperplane(hyperplane, **kwds):
r"""
Return the plot of a single hyperplane.
INPUT:
- ``**kwds`` -- plot options: see below
OUTPUT:
A graphics object of the plot.
.. RUBRIC:: Plot Options
Beside the usual plot options (enter ``plot?``), the plot command for
hyperplanes includes the following:
- ``hyperplane_label`` -- Boolean value or string (default: ``True``).
If ``True``, the hyperplane is labeled with its equation, if a
string, it is labeled by that string, otherwise it is not
labeled.
- ``label_color`` -- (Default: ``'black'``) Color for hyperplane_label.
- ``label_fontsize`` -- Size for ``hyperplane_label`` font (default: 14)
(does not work in 3d, yet).
- ``label_offset`` -- (Default: 0-dim: 0.1, 1-dim: (0,1),
2-dim: (0,0,0.2)) Amount by which label is offset from
``hyperplane.point()``.
- ``point_size`` -- (Default: 50) Size of points in a zero-dimensional
arrangement or of an arrangement over a finite field.
- ``ranges`` -- Range for the parameters for the parametric plot of the
hyperplane. If a single positive number ``r`` is given for the
value of ``ranges``, then the ranges for all parameters are set to
`[-r, r]`. Otherwise, for a line in the plane, ``ranges`` has the
form ``[a, b]`` (default: [-3,3]), and for a plane in 3-space, the
``ranges`` has the form ``[[a, b], [c, d]]`` (default: [[-3,3],[-3,3]]).
(The ranges are centered around ``hyperplane.point()``.)
EXAMPLES::
sage: H1.<x> = HyperplaneArrangements(QQ)
sage: a = 3*x + 4
sage: a.plot() # indirect doctest
Graphics object consisting of 3 graphics primitives
sage: a.plot(point_size=100,hyperplane_label='hello')
Graphics object consisting of 3 graphics primitives
sage: H2.<x,y> = HyperplaneArrangements(QQ)
sage: b = 3*x + 4*y + 5
sage: b.plot()
Graphics object consisting of 2 graphics primitives
sage: b.plot(ranges=(1,5),label_offset=(2,-1))
Graphics object consisting of 2 graphics primitives
sage: opts = {'hyperplane_label':True, 'label_color':'green',
....: 'label_fontsize':24, 'label_offset':(0,1.5)}
sage: b.plot(**opts)
Graphics object consisting of 2 graphics primitives
sage: H3.<x,y,z> = HyperplaneArrangements(QQ)
sage: c = 2*x + 3*y + 4*z + 5
sage: c.plot()
Graphics3d Object
sage: c.plot(label_offset=(1,0,1), color='green', label_color='red', frame=False)
Graphics3d Object
sage: d = -3*x + 2*y + 2*z + 3
sage: d.plot(opacity=0.8)
Graphics3d Object
sage: e = 4*x + 2*z + 3
sage: e.plot(ranges=[[-1,1],[0,8]], label_offset=(2,2,1), aspect_ratio=1)
Graphics3d Object
"""
if hyperplane.base_ring().characteristic():
raise NotImplementedError('base field must have characteristic zero')
elif hyperplane.dimension() not in [
0, 1, 2
]: # dimension of hyperplane, not ambient space
raise ValueError('can only plot hyperplanes in dimensions 1, 2, 3')
# handle extra keywords
if 'hyperplane_label' in kwds:
hyp_label = kwds.pop('hyperplane_label')
if not hyp_label:
has_hyp_label = False
else:
has_hyp_label = True
else: # default
hyp_label = True
has_hyp_label = True
if has_hyp_label:
if hyp_label: # then label hyperplane with its equation
if hyperplane.dimension() == 2: # jmol does not like latex
label = hyperplane._repr_linear(include_zero=False)
else:
label = hyperplane._latex_()
else:
label = hyp_label # a string
if 'label_color' in kwds:
label_color = kwds.pop('label_color')
else:
label_color = 'black'
if 'label_fontsize' in kwds:
label_fontsize = kwds.pop('label_fontsize')
else:
label_fontsize = 14
if 'label_offset' in kwds:
has_offset = True
label_offset = kwds.pop('label_offset')
else:
has_offset = False # give default values below
if 'point_size' in kwds:
pt_size = kwds.pop('point_size')
else:
pt_size = 50
if 'ranges' in kwds:
ranges_set = True
ranges = kwds.pop('ranges')
else:
ranges_set = False # give default values below
# the extra keywords have now been handled
# now create the plot
if hyperplane.dimension() == 0: # a point on a line
x, = hyperplane.A()
d = hyperplane.b()
p = point((d / x, 0), size=pt_size, **kwds)
if has_hyp_label:
if not has_offset:
label_offset = 0.1
p += text(label, (d / x, label_offset),
color=label_color,
fontsize=label_fontsize)
p += text('', (d / x, label_offset + 0.4)) # add space at top
if 'ymax' not in kwds:
kwds['ymax'] = 0.5
elif hyperplane.dimension() == 1: # a line in the plane
pnt = hyperplane.point()
w = hyperplane.linear_part().matrix()
t = SR.var('t')
if ranges_set:
if isinstance(ranges, (list, tuple)):
t0, t1 = ranges
else: # ranges should be a single positive number
t0, t1 = -ranges, ranges
else: # default
t0, t1 = -3, 3
p = parametric_plot(pnt + t * w[0], (t, t0, t1), **kwds)
if has_hyp_label:
if has_offset:
b0, b1 = label_offset
else:
b0, b1 = 0, 0.2
label = text(label, (pnt[0] + b0, pnt[1] + b1),
color=label_color,
fontsize=label_fontsize)
p += label
elif hyperplane.dimension() == 2: # a plane in 3-space
pnt = hyperplane.point()
w = hyperplane.linear_part().matrix()
s, t = SR.var('s t')
if ranges_set:
if isinstance(ranges, (list, tuple)):
s0, s1 = ranges[0]
t0, t1 = ranges[1]
else: # ranges should be a single positive integers
s0, s1 = -ranges, ranges
t0, t1 = -ranges, ranges
else: # default
s0, s1 = -3, 3
t0, t1 = -3, 3
p = parametric_plot3d(pnt + s * w[0] + t * w[1], (s, s0, s1),
(t, t0, t1), **kwds)
if has_hyp_label:
if has_offset:
b0, b1, b2 = label_offset
else:
b0, b1, b2 = 0, 0, 0
label = text3d(label, (pnt[0] + b0, pnt[1] + b1, pnt[2] + b2),
color=label_color,
fontsize=label_fontsize)
p += label
return p
def plot_hyperplane(hyperplane, **kwds):
r"""
Return the plot of a single hyperplane.
INPUT:
- ``**kwds`` -- plot options: see below
OUTPUT:
A graphics object of the plot.
.. RUBRIC:: Plot Options
Beside the usual plot options (enter ``plot?``), the plot command for
hyperplanes includes the following:
- ``hyperplane_label`` -- Boolean value or string (default: ``True``).
If ``True``, the hyperplane is labeled with its equation, if a
string, it is labeled by that string, otherwise it is not
labeled.
- ``label_color`` -- (Default: ``'black'``) Color for hyperplane_label.
- ``label_fontsize`` -- Size for ``hyperplane_label`` font (default: 14)
(does not work in 3d, yet).
- ``label_offset`` -- (Default: 0-dim: 0.1, 1-dim: (0,1),
2-dim: (0,0,0.2)) Amount by which label is offset from
``hyperplane.point()``.
- ``point_size`` -- (Default: 50) Size of points in a zero-dimensional
arrangement or of an arrangement over a finite field.
- ``ranges`` -- Range for the parameters for the parametric plot of the
hyperplane. If a single positive number ``r`` is given for the
value of ``ranges``, then the ranges for all parameters are set to
`[-r, r]`. Otherwise, for a line in the plane, ``ranges`` has the
form ``[a, b]`` (default: [-3,3]), and for a plane in 3-space, the
``ranges`` has the form ``[[a, b], [c, d]]`` (default: [[-3,3],[-3,3]]).
(The ranges are centered around ``hyperplane.point()``.)
EXAMPLES::
sage: H1.<x> = HyperplaneArrangements(QQ)
sage: a = 3*x + 4
sage: a.plot() # indirect doctest
Graphics object consisting of 3 graphics primitives
sage: a.plot(point_size=100,hyperplane_label='hello')
Graphics object consisting of 3 graphics primitives
sage: H2.<x,y> = HyperplaneArrangements(QQ)
sage: b = 3*x + 4*y + 5
sage: b.plot()
Graphics object consisting of 2 graphics primitives
sage: b.plot(ranges=(1,5),label_offset=(2,-1))
Graphics object consisting of 2 graphics primitives
sage: opts = {'hyperplane_label':True, 'label_color':'green',
....: 'label_fontsize':24, 'label_offset':(0,1.5)}
sage: b.plot(**opts)
Graphics object consisting of 2 graphics primitives
sage: H3.<x,y,z> = HyperplaneArrangements(QQ)
sage: c = 2*x + 3*y + 4*z + 5
sage: c.plot()
Graphics3d Object
sage: c.plot(label_offset=(1,0,1), color='green', label_color='red', frame=False)
Graphics3d Object
sage: d = -3*x + 2*y + 2*z + 3
sage: d.plot(opacity=0.8)
Graphics3d Object
sage: e = 4*x + 2*z + 3
sage: e.plot(ranges=[[-1,1],[0,8]], label_offset=(2,2,1), aspect_ratio=1)
Graphics3d Object
"""
if hyperplane.base_ring().characteristic() != 0:
raise NotImplementedError('base field must have characteristic zero')
elif hyperplane.dimension() not in [0, 1, 2]: # dimension of hyperplane, not ambient space
raise ValueError('can only plot hyperplanes in dimensions 1, 2, 3')
# handle extra keywords
if 'hyperplane_label' in kwds:
hyp_label = kwds.pop('hyperplane_label')
if not hyp_label:
has_hyp_label = False
else:
has_hyp_label = True
else: # default
hyp_label = True
has_hyp_label = True
if has_hyp_label:
if hyp_label: # then label hyperplane with its equation
if hyperplane.dimension() == 2: # jmol does not like latex
label = hyperplane._repr_linear(include_zero=False)
else:
label = hyperplane._latex_()
else:
label = hyp_label # a string
if 'label_color' in kwds:
label_color = kwds.pop('label_color')
else:
label_color = 'black'
if 'label_fontsize' in kwds:
label_fontsize = kwds.pop('label_fontsize')
else:
label_fontsize = 14
if 'label_offset' in kwds:
has_offset = True
label_offset = kwds.pop('label_offset')
else:
has_offset = False # give default values below
if 'point_size' in kwds:
pt_size = kwds.pop('point_size')
else:
pt_size = 50
if 'ranges' in kwds:
ranges_set = True
ranges = kwds.pop('ranges')
else:
ranges_set = False # give default values below
# the extra keywords have now been handled
# now create the plot
if hyperplane.dimension() == 0: # a point on a line
x, = hyperplane.A()
d = hyperplane.b()
p = point((d/x,0), size = pt_size, **kwds)
if has_hyp_label:
if not has_offset:
label_offset = 0.1
p += text(label, (d/x,label_offset),
color=label_color,fontsize=label_fontsize)
p += text('',(d/x,label_offset+0.4)) # add space at top
if 'ymax' not in kwds:
kwds['ymax'] = 0.5
elif hyperplane.dimension() == 1: # a line in the plane
pnt = hyperplane.point()
w = hyperplane.linear_part().matrix()
x, y = hyperplane.A()
d = hyperplane.b()
t = SR.var('t')
if ranges_set:
if type(ranges) in [list,tuple]:
t0, t1 = ranges
else: # ranges should be a single positive number
t0, t1 = -ranges, ranges
else: # default
t0, t1 = -3, 3
p = parametric_plot(pnt+t*w[0], (t,t0,t1), **kwds)
if has_hyp_label:
if has_offset:
b0, b1 = label_offset
else:
b0, b1 = 0, 0.2
label = text(label,(pnt[0]+b0,pnt[1]+b1),
color=label_color,fontsize=label_fontsize)
p += label
elif hyperplane.dimension() == 2: # a plane in 3-space
pnt = hyperplane.point()
w = hyperplane.linear_part().matrix()
a, b, c = hyperplane.A()
d = hyperplane.b()
s,t = SR.var('s t')
if ranges_set:
if type(ranges) in [list,tuple]:
s0, s1 = ranges[0]
t0, t1 = ranges[1]
else: # ranges should be a single positive integers
s0, s1 = -ranges, ranges
t0, t1 = -ranges, ranges
else: # default
s0, s1 = -3, 3
t0, t1 = -3, 3
p = parametric_plot3d(pnt+s*w[0]+t*w[1],(s,s0,s1),(t,t0,t1),**kwds)
if has_hyp_label:
if has_offset:
b0, b1, b2 = label_offset
else:
b0, b1, b2 = 0, 0, 0
label = text3d(label,(pnt[0]+b0,pnt[1]+b1,pnt[2]+b2),
color=label_color,fontsize=label_fontsize)
p += label
return p
radius=norm(first_vector)
if norm(second_vector)!=radius:
raise ValueError("Ellipse not implemented")
first_unit_vector=first_vector/radius
second_unit_vector=second_vector/radius
normal_vector=second_vector-(second_vector*first_unit_vector)*first_unit_vector
if norm(normal_vector)==0:
print (first_point,second_point)
return
normal_unit_vector=normal_vector/norm(normal_vector)
scalar_product=first_unit_vector*second_unit_vector
if abs(scalar_product) == 1:
raise ValueError("The points are alligned")
angle=arccos(scalar_product)
var('t')
return parametric_plot3d(center+first_vector*cos(t)+radius*normal_unit_vector*sin(t),(0,angle),**kwds)
def _arc(p,q,s,**kwds):
#rewrite this to use polar_plot and get points to do filled triangles
from sage.misc.functional import det
from sage.plot.line import line
from sage.misc.functional import norm
from sage.symbolic.all import pi
from sage.plot.arc import arc
p,q,s = map( lambda x: vector(x), [p,q,s])
# to avoid running into division by 0 we set to be colinear vectors that are
# almost colinear
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)
raise ValueError("Ellipse not implemented")
first_unit_vector = first_vector / radius
second_unit_vector = second_vector / radius
normal_vector = second_vector - (second_vector *
first_unit_vector) * first_unit_vector
if norm(normal_vector) == 0:
print(first_point, second_point)
return
normal_unit_vector = normal_vector / norm(normal_vector)
scalar_product = first_unit_vector * second_unit_vector
if abs(scalar_product) == 1:
raise ValueError("The points are alligned")
angle = arccos(scalar_product)
var('t')
return parametric_plot3d(
center + first_vector * cos(t) + radius * normal_unit_vector * sin(t),
(0, angle), **kwds)
def _arc(p, q, s, **kwds):
#rewrite this to use polar_plot and get points to do filled triangles
from sage.misc.functional import det
from sage.plot.line import line
from sage.misc.functional import norm
from sage.symbolic.all import pi
from sage.plot.arc import arc
p, q, s = map(lambda x: vector(x), [p, q, s])
# to avoid running into division by 0 we set to be colinear vectors that are
# almost colinear
