Surfaces of revolution

AUTHORS:

  • Oscar Gerardo Lazo Arjona (2010): initial version.
sage.plot.plot3d.revolution_plot3d.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))
../../../_images/revolution_plot3d-1.svg

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))
../../../_images/revolution_plot3d-2.svg

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()
../../../_images/revolution_plot3d-3.svg

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))
../../../_images/revolution_plot3d-4.svg

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))
../../../_images/revolution_plot3d-5.svg

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))
../../../_images/revolution_plot3d-6.svg

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))
../../../_images/revolution_plot3d-7.svg

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)
../../../_images/revolution_plot3d-8.svg

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))
../../../_images/revolution_plot3d-9.svg

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)
../../../_images/revolution_plot3d-10.svg