Surfaces of revolution


  • 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.


  • 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.


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)

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