The current undesirable behavior of plot3d

var("x y z")
plot3d( x^2 + y^3, (x,-3,3), (y,-3,3) )

(x, y, z)

3D rendering not yet implemented

var("x y")
plot3d( 1/( (x-2)^2 + (y-1)^2 ), (x,-3,3), (y,-3,3) )

(x, y)

3D rendering not yet implemented

Now, the behavior of the new code that I've written

%load new_plot3d_syntax.sage

newPlot3d( x^2 + y^3, -3,3, -3,3, -4,4 )

<string>:1: DeprecationWarning: Substitution using function-call syntax and unnamed arguments is deprecated and will be removed from a future release of Sage; you can use named arguments instead, like EXPR(x=..., y=...)
See http://trac.sagemath.org/5930 for details.

3D rendering not yet implemented

newPlot3d( 1/( (x-2)^2 + (y-1)^2 ), -3,3, -3,3, 0,6 )

3D rendering not yet implemented

What are some of the changes?

Functions with vertical asymptotes were not graphable at all before
- Before, you'd get something that looks like a vertical pencil.
- Now they can be graphed like any other function.
Before, you could not bound the desired z-coordinates at all
- Now such bounds are obligatory.
- This is what allows for functions with vertical asymptotes.
Before, the aspect ratio defaults to [1,1,1]
- Now, the aspect ratio is computed for the user so as to produce a graph in a cube.
- Many images, in the old code, were hard to interpret/understand because of the aspect ratios.
- The aspect ratio requires knowledge of the bounds of the function.
  - I don't think a student at the start of learning about 3D functions can compute those bounds unassisted.
  - Yet, without those bounds, the aspect ratio to produce a cube cannot be computed.
  - And without the aspect ratios being provided by the user in the old code, the image is unreadable.
  - Therefore, with the current commands, newbie users cannot just sit down and graph in 3D.
- You can always override the default aspect ratios, in either case, by an option.
Before the mesh defaults to "off", but you can turn it on with an option.
- Now the mesh defaults to "on", but you can turn it off with an option.

sage: from sage.plot.plot3d.implicit_surface import ImplicitSurface
%var x,y,z
contour=[-1, 0, 1, 2]
from numpy import linspace
where = linspace(min(contour), max(contour), len(contour)) / (max(contour) - min(contour))
where -= where[0]
colors = [colormaps.coolwarm(i) for i in where]
sum(implicit_plot3d( 1/( (x-2)^2 + (y-1)^2 ) == z, (x,-3, 3), (y,-3, 3), (z, -1, 6), contour=con, color=col, mesh=True) for con, col in zip(contour, colors))

3D rendering not yet implemented

implicit_plot3d(  1/( (x-2)^2 + (y-1)^2 ) == z, (x,-3, 3), (y,-3, 3), (z, 0, 6), mesh=True, color='green')

3D rendering not yet implemented