Coverage for local/lib/python2.7/site-packages/sage/plot/plot3d/revolution_plot3d.py : 74%

""" 

Surfaces of revolution 

AUTHORS: 

- Oscar Gerardo Lazo Arjona (2010): initial version. 

""" 

#***************************************************************************** 

# Copyright (C) 2010 Oscar Gerardo Lazo Arjona algebraicamente@gmail.com 

# 

# Distributed under the terms of the GNU General Public License (GPL) 

# 

# This code is distributed in the hope that it will be useful, 

# but WITHOUT ANY WARRANTY; without even the implied warranty of 

# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU 

# General Public License for more details. 

# 

# The full text of the GPL is available at: 

# 

# http://www.gnu.org/licenses/ 

#***************************************************************************** 

from __future__ import print_function 

from sage.plot.plot3d.parametric_plot3d import parametric_plot3d 

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)) 

.. 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:: 

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). 

""" 

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, 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)