parametric_plot

[Ricardo Acuna]

Hello,

I'm trying to recreate a plot that I did in Maple 

P := (x, y) -> WeierstrassP(x + y*I, 1, 0);

PP := (x, y) -> WeierstrassPPrime(x + y*I, 1, 0);

IR := (theta, z) -> cos(theta)*Re(z) + sin(theta)*Im(z);

Gr1 := theta -> [[Re(P(x, y)), Im(P(x, y)), 0.3*IR(theta, PP(x, y))], x = 0.001 .. 3.74, y = 0.001 .. 3.74, color = [sin(2*Pi*x/3.74), 0.5, sin(2*Pi*y/3.74)], view = [-1 .. 1, -1 .. 1, -1 .. 1], grid = [35, 35]];

plots[animate](plot3d, Gr1(t), t = 0 .. Pi - 0.1)

Here's what I have so far.

var('z')

E = EllipticCurve(QQ, [0,0,0,-1/4,0])

wp = E.weierstrass_p().laurent_polynomial()

wpp = derivative(wp,z)

f = (lambda u,v: wp(u+i*v).real(), lambda u,v: wp(u+i*v).imag(), lambda u,v: wpp(u+i*v).imag())

parametric_plot3d(f, (0.001,3.74), (0.001,3.74))

the only problem I see is that I have no control over the view box. With Maple I can have the parameter view = [-1 .. 1, -1 .. 1, -1 .. 1] such that the software only shows stuff in that range.

contour_plot has the optional region parameter, so it's possible someone already implemented something similar in another class.

Best,

RJ Acuña

[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)

```

[slelievre]

I was wrong! The functionality exists!

Good thing you asked the question again as

- Ask Sage question 58169

  Change the view box on a 3D graph

  https://ask.sagemath.org/question/58169

and Frédéric Chapoton answered it there,

pointing to the `add_condition` method!

Remember to accept his answer to mark your

question as solved in the list of questions.

Please ignore my previous previous post and

the less than satisfactory approach in it.