[sage-support] Re: parametric_plot

[slelievre]

So far, limiting the z range in 3D plots from plot3d,
parametric_plot or parametric_plot3d is missing.

Providing this feature is tracked at

- Sage Trac ticket 31264
  Allow setting zmin, zmax in plot3d and other 3d plots
  https://trac.sagemath.org/ticket/31264

Using implicit_plot3d can be a workaround,
given an equation for the surface ...which is easy
in the case of plot3d but usually not in the case
of parametric_plot3d.

To limit the x, y and z ranges in a parametric plot,
we can alternatively use auxiliary functions which
return their computed value when it is within bounds,
and return "not a number" otherwise.

Here is a way to do that for the plot in the question.
We use cached functions to save computation time.

```
z = SR.var('z')
E = EllipticCurve(QQ, [0, 0, 0, -1/4, 0])
wp = E.weierstrass_p().laurent_polynomial()
wpp = derivative(wp, z)

nan = float('nan')

@cached_function
def wpuv(u, v):
    return wp(u + i*v)

@cached_function
def xuv(u, v):
    return wpuv(u, v).real()

@cached_function
def yuv(u, v):
    return wpuv(u, v).imag()

@cached_function
def zuv(u, v):
    return wpp(u + i*v).imag()

xmin, xmax = -1, 1
ymin, ymax = -1, 1
zmin, zmax = -1, 1

@cached_function
def xok(u, v):
    return xmin < xuv(u, v) < xmax

@cached_function
def yok(u, v):
    return ymin < yuv(u, v) < ymax

@cached_function
def zok(u, v):
    return zmin < zuv(u, v) < zmax

@cached_function
def xyzok(u, v):
    return xok(u, v) and yok(u, v) and zok(u, v)

@cached_function
def xxuv(u, v):
    return xuv(u, v) if xyzok(u, v) else nan

@cached_function
def yyuv(u, v):
    return yuv(u, v) if xyzok(u, v) else nan

@cached_function
def zzuv(u, v):
    return zuv(u, v) if xyzok(u, v) else nan

uu = (1e-3, 3.74)
vv = (1e-3, 3.74)

pp = parametric_plot3d

pp([xxuv, yyuv, zzuv], uu, vv)
```

That's already an interesting view.

To refine it, check which values of (u, v) give (x, y, z) within bounds:

```
good_uv_x = region_plot(xok, uu, vv)
good_uv_y = region_plot(yok, uu, vv)
good_uv_z = region_plot(zok, uu, vv)
good_uv_xyz = region_plot(xyzok, uu, vv)
good_uv = [good_uv_x, good_uv_y, good_uv_z, good_uv_xyz]

graphics_array(good_uv, ncols=2)
```

This reveals that values of u and v beyond 3.2 are not so useful.

Use a reduced range for u and v, and increase the number of plot points:

```
uu = (RDF(1e-3), RDF(3.2))
vv = (RDF(1e-3), RDF(3.2))
pp([xxuv, yyuv, zzuv], uu, vv, plot_points=129)
```

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