2017-12-18-215012.sagews

Demo de càlcul de varietats parametritzades

Primer exemple

Creem l'anell de polinomis amb el corresponent ordre monomial

A.<T,U,V,X,Y,Z> = PolynomialRing(QQ, order='lex')
A

Multivariate Polynomial Ring in T, U, V, X, Y, Z over Rational Field

Considerem la varietat parametritzada per $$\begin{aligned} X &= 3U+3UV^2-U^3\\ Y &= 3V+3U^2V-V^3\\ Z &= 3U^2-3V^2\\ \end{aligned}$$

Fem el dibuix d'aquesta parametrització

parametric_plot3d?

File: /ext/sage/sage-8.1/local/lib/python2.7/site-packages/sage/plot/plot3d/parametric_plot3d.py
Signature : parametric_plot3d(f, urange, vrange=None, plot_points='automatic', boundary_style=None, **kwds)
Docstring :
Return a parametric three-dimensional space curve or surface.

There are four ways to call this function:

* "parametric_plot3d([f_x, f_y, f_z], (u_min, u_max))": f_x, f_y,
  f_z are three functions and u_{min} and u_{max} are real
  numbers

* "parametric_plot3d([f_x, f_y, f_z], (u, u_min, u_max))": f_x,
  f_y, f_z can be viewed as functions of u

* "parametric_plot3d([f_x, f_y, f_z], (u_min, u_max), (v_min,
  v_max))": f_x, f_y, f_z are each functions of two variables

* "parametric_plot3d([f_x, f_y, f_z], (u, u_min, u_max), (v,
  v_min, v_max))": f_x, f_y, f_z can be viewed as functions of u
  and v

INPUT:

* "f" - a 3-tuple of functions or expressions, or vector of size
  3

* "urange" - a 2-tuple (u_min, u_max) or a 3-tuple (u, u_min,
  u_max)

* "vrange" - (optional - only used for surfaces) a 2-tuple
  (v_min, v_max) or a 3-tuple (v, v_min, v_max)

* "plot_points" - (default: "automatic", which is 75 for curves
  and [40,40] for surfaces) initial number of sample points in each
  parameter; an integer for a curve, and a pair of integers for a
  surface.

* "boundary_style" - (default: None, no boundary) a dict that
  describes how to draw the boundaries of regions by giving options
  that are passed to the line3d command.

* "mesh" - bool (default: False) whether to display mesh grid
  lines

* "dots" - bool (default: False) whether to display dots at mesh
  grid points

Note:

  1. By default for a curve any points where f_x, f_y, or f_z do
     not evaluate to a real number are skipped.

  2. Currently for a surface f_x, f_y, and f_z have to be
     defined everywhere. This will change.

  3. mesh and dots are not supported when using the Tachyon ray
     tracer renderer.

EXAMPLES: We demonstrate each of the four ways to call this
function.

1. A space curve defined by three functions of 1 variable:

         sage: parametric_plot3d((sin, cos, lambda u: u/10), (0,20))
         Graphics3d Object

   Note above the lambda function, which creates a callable Python
   function that sends u to u/10.

2. Next we draw the same plot as above, but using symbolic
   functions:

         sage: u = var('u')
         sage: parametric_plot3d((sin(u), cos(u), u/10), (u,0,20))
         Graphics3d Object

3. We draw a parametric surface using 3 Python functions
   (defined using lambda):

         sage: f = (lambda u,v: cos(u), lambda u,v: sin(u)+cos(v), lambda u,v: sin(v))
         sage: parametric_plot3d(f, (0,2*pi), (-pi,pi))
         Graphics3d Object

4. The same surface, but where the defining functions are
   symbolic:

         sage: u, v = var('u,v')
         sage: parametric_plot3d((cos(u), sin(u)+cos(v), sin(v)), (u,0,2*pi), (v,-pi,pi))
         Graphics3d Object

The surface, but with a mesh:

   sage: u, v = var('u,v')
   sage: parametric_plot3d((cos(u), sin(u)+cos(v), sin(v)), (u,0,2*pi), (v,-pi,pi), mesh=True)
   Graphics3d Object

We increase the number of plot points, and make the surface green
and transparent:

   sage: parametric_plot3d((cos(u), sin(u)+cos(v), sin(v)), (u,0,2*pi), (v,-pi,pi),
   ....: color='green', opacity=0.1, plot_points=[30,30])
   Graphics3d Object

One can also color the surface using a coloring function and a
colormap:

   sage: u,v = var('u,v')
   sage: def cf(u,v): return sin(u+v/2)**2
   sage: P = parametric_plot3d((cos(u), sin(u)+cos(v), sin(v)),
   ....:   (u,0,2*pi), (v,-pi,pi), color=(cf,colormaps.PiYG), plot_points=[60,60])
   sage: P.show(viewer='tachyon')

Another example, a colored Möbius band:

   sage: cm = colormaps.ocean
   sage: def c(x,y): return sin(x*y)**2
   sage: from sage.plot.plot3d.parametric_surface import MoebiusStrip
   sage: MoebiusStrip(5, 1, plot_points=200, color=(c,cm))
   Graphics3d Object

Yet another colored example:

   sage: from sage.plot.plot3d.parametric_surface import ParametricSurface
   sage: cm = colormaps.autumn
   sage: def c(x,y): return sin(x*y)**2
   sage: def g(x,y): return x, y+sin(y), x**2 + y**2
   sage: ParametricSurface(g, (srange(-10,10,0.1), srange(-5,5.0,0.1)), color=(c,cm))
   Graphics3d Object

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).

We call the space curve function but with polynomials instead of
symbolic variables.

   sage: R.<t> = RDF[]
   sage: parametric_plot3d((t, t^2, t^3), (t,0,3))
   Graphics3d Object

Next we plot the same curve, but because we use (0, 3) instead of
(t, 0, 3), each polynomial is viewed as a callable function of one
variable:

   sage: parametric_plot3d((t, t^2, t^3), (0,3))
   Graphics3d Object

We do a plot but mix a symbolic input, and an integer:

   sage: t = var('t')
   sage: parametric_plot3d((1, sin(t), cos(t)), (t,0,3))
   Graphics3d Object

We specify a boundary style to show us the values of the function
at its extrema:

   sage: u, v = var('u,v')
   sage: parametric_plot3d((cos(u), sin(u)+cos(v), sin(v)), (u,0,pi), (v,0,pi),
   ....:              boundary_style={"color": "black", "thickness": 2})
   Graphics3d Object

We can plot vectors:

   sage: x,y = var('x,y')
   sage: parametric_plot3d(vector([x-y, x*y, x*cos(y)]), (x,0,2), (y,0,2))
   Graphics3d Object

   sage: t = var('t')
   sage: p = vector([1,2,3])
   sage: q = vector([2,-1,2])
   sage: parametric_plot3d(p*t+q, (t,0,2))
   Graphics3d Object

Any options you would normally use to specify the appearance of a
curve are valid as entries in the "boundary_style" dict.

MANY MORE EXAMPLES:

We plot two interlinked tori:

   sage: u, v = var('u,v')
   sage: f1 = (4+(3+cos(v))*sin(u), 4+(3+cos(v))*cos(u), 4+sin(v))
   sage: f2 = (8+(3+cos(v))*cos(u), 3+sin(v), 4+(3+cos(v))*sin(u))
   sage: p1 = parametric_plot3d(f1, (u,0,2*pi), (v,0,2*pi), texture="red")
   sage: p2 = parametric_plot3d(f2, (u,0,2*pi), (v,0,2*pi), texture="blue")
   sage: p1 + p2
   Graphics3d Object

A cylindrical Star of David:

   sage: u,v = var('u v')
   sage: K = (abs(cos(u))^200+abs(sin(u))^200)^(-1.0/200)
   sage: f_x = cos(u) * cos(v) * (abs(cos(3*v/4))^500+abs(sin(3*v/4))^500)^(-1/260) * K
   sage: f_y = cos(u) * sin(v) * (abs(cos(3*v/4))^500+abs(sin(3*v/4))^500)^(-1/260) * K
   sage: f_z = sin(u) * K
   sage: parametric_plot3d([f_x, f_y, f_z], (u, -pi, pi), (v, 0, 2*pi))
   Graphics3d Object

Double heart:

   sage: u, v = var('u,v')
   sage: G1 = abs(sqrt(2)*tanh((u/sqrt(2))))
   sage: G2 = abs(sqrt(2)*tanh((v/sqrt(2))))
   sage: f_x = (abs(v) - abs(u) - G1 + G2)*sin(v)
   sage: f_y = (abs(v) - abs(u) - G1 - G2)*cos(v)
   sage: f_z = sin(u)*(abs(cos(u)) + abs(sin(u)))^(-1)
   sage: parametric_plot3d([f_x, f_y, f_z], (u,0,pi), (v,-pi,pi))
   Graphics3d Object

Heart:

   sage: u, v = var('u,v')
   sage: f_x = cos(u)*(4*sqrt(1-v^2)*sin(abs(u))^abs(u))
   sage: f_y = sin(u)*(4*sqrt(1-v^2)*sin(abs(u))^abs(u))
   sage: f_z = v
   sage: parametric_plot3d([f_x, f_y, f_z], (u,-pi,pi), (v,-1,1), frame=False, color="red")
   Graphics3d Object

A Trefoil knot https://en.wikipedia.org/wiki/Trefoil_knot:

   sage: u, v = var('u,v')
   sage: f_x = (4*(1+0.25*sin(3*v))+cos(u))*cos(2*v)
   sage: f_y = (4*(1+0.25*sin(3*v))+cos(u))*sin(2*v)
   sage: f_z = sin(u)+2*cos(3*v)
   sage: parametric_plot3d([f_x, f_y, f_z], (u,-pi,pi), (v,-pi,pi), frame=False, color="blue")
   Graphics3d Object

Green bowtie:

   sage: u, v = var('u,v')
   sage: f_x = sin(u) / (sqrt(2) + sin(v))
   sage: f_y = sin(u) / (sqrt(2) + cos(v))
   sage: f_z = cos(u) / (1 + sqrt(2))
   sage: parametric_plot3d([f_x, f_y, f_z], (u,-pi,pi), (v,-pi,pi), frame=False, color="green")
   Graphics3d Object

Boy's surface http://en.wikipedia.org/wiki/Boy's_surface and
http://mathcurve.com/surfaces/boy/boy.shtml:

   sage: u, v = var('u,v')
   sage: K = cos(u) / (sqrt(2) - cos(2*u)*sin(3*v))
   sage: f_x = K * (cos(u)*cos(2*v)+sqrt(2)*sin(u)*cos(v))
   sage: f_y = K * (cos(u)*sin(2*v)-sqrt(2)*sin(u)*sin(v))
   sage: f_z = 3 * K * cos(u)
   sage: parametric_plot3d([f_x, f_y, f_z], (u,-2*pi,2*pi), (v,0,pi),
   ....:                   plot_points=[90,90], frame=False, color="orange") # long time -- about 30 seconds
   Graphics3d Object

Maeder's Owl also known as Bour's minimal surface
https://en.wikipedia.org/wiki/Bour%27s_minimal_surface:

   sage: u, v = var('u,v')
   sage: f_x = v*cos(u) - 0.5*v^2*cos(2*u)
   sage: f_y = -v*sin(u) - 0.5*v^2*sin(2*u)
   sage: f_z = 4 * v^1.5 * cos(3*u/2) / 3
   sage: parametric_plot3d([f_x, f_y, f_z], (u,-2*pi,2*pi), (v,0,1),
   ....:                    plot_points=[90,90], frame=False, color="purple")
   Graphics3d Object

Bracelet:

   sage: u, v = var('u,v')
   sage: f_x = (2 + 0.2*sin(2*pi*u))*sin(pi*v)
   sage: f_y = 0.2 * cos(2*pi*u) * 3 * cos(2*pi*v)
   sage: f_z = (2 + 0.2*sin(2*pi*u))*cos(pi*v)
   sage: parametric_plot3d([f_x, f_y, f_z], (u,0,pi/2), (v,0,3*pi/4), frame=False, color="gray")
   Graphics3d Object

Green goblet:

   sage: u, v = var('u,v')
   sage: f_x = cos(u) * cos(2*v)
   sage: f_y = sin(u) * cos(2*v)
   sage: f_z = sin(v)
   sage: parametric_plot3d([f_x, f_y, f_z], (u,0,2*pi), (v,0,pi), frame=False, color="green")
   Graphics3d Object

Funny folded surface - with square projection:

   sage: u, v = var('u,v')
   sage: f_x = cos(u) * sin(2*v)
   sage: f_y = sin(u) * cos(2*v)
   sage: f_z = sin(v)
   sage: parametric_plot3d([f_x, f_y, f_z], (u,0,2*pi), (v,0,2*pi), frame=False, color="green")
   Graphics3d Object

Surface of revolution of figure 8:

   sage: u, v = var('u,v')
   sage: f_x = cos(u) * sin(2*v)
   sage: f_y = sin(u) * sin(2*v)
   sage: f_z = sin(v)
   sage: parametric_plot3d([f_x, f_y, f_z], (u,0,2*pi), (v,0,2*pi), frame=False, color="green")
   Graphics3d Object

Yellow Whitney's umbrella
http://en.wikipedia.org/wiki/Whitney_umbrella:

   sage: u, v = var('u,v')
   sage: f_x = u*v
   sage: f_y = u
   sage: f_z = v^2
   sage: parametric_plot3d([f_x, f_y, f_z], (u,-1,1), (v,-1,1), frame=False, color="yellow")
   Graphics3d Object

Cross cap http://en.wikipedia.org/wiki/Cross-cap:

   sage: u, v = var('u,v')
   sage: f_x = (1+cos(v)) * cos(u)
   sage: f_y = (1+cos(v)) * sin(u)
   sage: f_z = -tanh((2/3)*(u-pi)) * sin(v)
   sage: parametric_plot3d([f_x, f_y, f_z], (u,0,2*pi), (v,0,2*pi), frame=False, color="red")
   Graphics3d Object

Twisted torus:

   sage: u, v = var('u,v')
   sage: f_x = (3+sin(v)+cos(u)) * cos(2*v)
   sage: f_y = (3+sin(v)+cos(u)) * sin(2*v)
   sage: f_z = sin(u) + 2*cos(v)
   sage: parametric_plot3d([f_x, f_y, f_z], (u,0,2*pi), (v,0,2*pi), frame=False, color="red")
   Graphics3d Object

Four intersecting discs:

   sage: u, v = var('u,v')
   sage: f_x = v*cos(u) - 0.5*v^2*cos(2*u)
   sage: f_y = -v*sin(u) - 0.5*v^2*sin(2*u)
   sage: f_z = 4 * v^1.5 * cos(3*u/2) / 3
   sage: parametric_plot3d([f_x, f_y, f_z], (u,0,4*pi), (v,0,2*pi), frame=False, color="red", opacity=0.7)
   Graphics3d Object

Steiner surface/Roman's surface (see
http://en.wikipedia.org/wiki/Roman_surface and
http://en.wikipedia.org/wiki/Steiner_surface):

   sage: u, v = var('u,v')
   sage: f_x = (sin(2*u) * cos(v) * cos(v))
   sage: f_y = (sin(u) * sin(2*v))
   sage: f_z = (cos(u) * sin(2*v))
   sage: parametric_plot3d([f_x, f_y, f_z], (u,-pi/2,pi/2), (v,-pi/2,pi/2), frame=False, color="red")
   Graphics3d Object

Klein bottle? (see http://en.wikipedia.org/wiki/Klein_bottle):

   sage: u, v = var('u,v')
   sage: f_x = (3*(1+sin(v)) + 2*(1-cos(v)/2)*cos(u)) * cos(v)
   sage: f_y = (4+2*(1-cos(v)/2)*cos(u)) * sin(v)
   sage: f_z = -2 * (1-cos(v)/2) * sin(u)
   sage: parametric_plot3d([f_x, f_y, f_z], (u,0,2*pi), (v,0,2*pi), frame=False, color="green")
   Graphics3d Object

A Figure 8 embedding of the Klein bottle (see
http://en.wikipedia.org/wiki/Klein_bottle):

   sage: u, v = var('u,v')
   sage: f_x = (2+cos(v/2)*sin(u)-sin(v/2)*sin(2*u)) * cos(v)
   sage: f_y = (2+cos(v/2)*sin(u)-sin(v/2)*sin(2*u)) * sin(v)
   sage: f_z = sin(v/2)*sin(u) + cos(v/2)*sin(2*u)
   sage: parametric_plot3d([f_x, f_y, f_z], (u,0,2*pi), (v,0,2*pi), frame=False, color="red")
   Graphics3d Object

Enneper's surface (see
http://en.wikipedia.org/wiki/Enneper_surface):

   sage: u, v = var('u,v')
   sage: f_x = u - u^3/3 + u*v^2
   sage: f_y = v - v^3/3 + v*u^2
   sage: f_z = u^2 - v^2
   sage: parametric_plot3d([f_x, f_y, f_z], (u,-2,2), (v,-2,2), frame=False, color="red")
   Graphics3d Object

Henneberg's surface (see http://xahlee.org/surface/gallery_m.html):

   sage: u, v = var('u,v')
   sage: f_x = 2*sinh(u)*cos(v) - (2/3)*sinh(3*u)*cos(3*v)
   sage: f_y = 2*sinh(u)*sin(v) + (2/3)*sinh(3*u)*sin(3*v)
   sage: f_z = 2 * cosh(2*u) * cos(2*v)
   sage: parametric_plot3d([f_x, f_y, f_z], (u,-1,1), (v,-pi/2,pi/2), frame=False, color="red")
   Graphics3d Object

Dini's spiral:

   sage: u, v = var('u,v')
   sage: f_x = cos(u) * sin(v)
   sage: f_y = sin(u) * sin(v)
   sage: f_z = (cos(v)+log(tan(v/2))) + 0.2*u
   sage: parametric_plot3d([f_x, f_y, f_z], (u,0,12.4), (v,0.1,2), frame=False, color="red")
   Graphics3d Object

Catalan's surface (see
http://xahlee.org/surface/catalan/catalan.html):

   sage: u, v = var('u,v')
   sage: f_x = u - sin(u)*cosh(v)
   sage: f_y = 1 - cos(u)*cosh(v)
   sage: f_z = 4 * sin(1/2*u) * sinh(v/2)
   sage: parametric_plot3d([f_x, f_y, f_z], (u,-pi,3*pi), (v,-2,2), frame=False, color="red")
   Graphics3d Object

A Conchoid:

   sage: u, v = var('u,v')
   sage: k = 1.2; k_2 = 1.2; a = 1.5
   sage: f = (k^u*(1+cos(v))*cos(u), k^u*(1+cos(v))*sin(u), k^u*sin(v)-a*k_2^u)
   sage: parametric_plot3d(f, (u,0,6*pi), (v,0,2*pi), plot_points=[40,40], texture=(0,0.5,0))
   Graphics3d Object

A Möbius strip:

   sage: u,v = var("u,v")
   sage: parametric_plot3d([cos(u)*(1+v*cos(u/2)), sin(u)*(1+v*cos(u/2)), 0.2*v*sin(u/2)],
   ....:                   (u,0, 4*pi+0.5), (v,0, 0.3), plot_points=[50,50])
   Graphics3d Object

A Twisted Ribbon:

   sage: u, v = var('u,v')
   sage: parametric_plot3d([3*sin(u)*cos(v), 3*sin(u)*sin(v), cos(v)],
   ....:                   (u,0,2*pi), (v,0,pi), plot_points=[50,50])
   Graphics3d Object

An Ellipsoid:

   sage: u, v = var('u,v')
   sage: parametric_plot3d([3*sin(u)*cos(v), 2*sin(u)*sin(v), cos(u)],
   ....:                   (u,0, 2*pi), (v, 0, 2*pi), plot_points=[50,50], aspect_ratio=[1,1,1])
   Graphics3d Object

A Cone:

   sage: u, v = var('u,v')
   sage: parametric_plot3d([u*cos(v), u*sin(v), u], (u,-1,1), (v,0,2*pi+0.5), plot_points=[50,50])
   Graphics3d Object

A Paraboloid:

   sage: u, v = var('u,v')
   sage: parametric_plot3d([u*cos(v), u*sin(v), u^2], (u,0,1), (v,0,2*pi+0.4), plot_points=[50,50])
   Graphics3d Object

A Hyperboloid:

   sage: u, v = var('u,v')
   sage: plot3d(u^2-v^2, (u,-1,1), (v,-1,1), plot_points=[50,50])
   Graphics3d Object

A weird looking surface - like a Möbius band but also an O:

   sage: u, v = var('u,v')
   sage: parametric_plot3d([sin(u)*cos(u)*log(u^2)*sin(v), (u^2)^(1/6)*(cos(u)^2)^(1/4)*cos(v), sin(v)],
   ....:                   (u,0.001,1), (v,-pi,pi+0.2), plot_points=[50,50])
   Graphics3d Object

A heart, but not a cardioid (for my wife):

   sage: u, v = var('u,v')
   sage: p1 = parametric_plot3d([sin(u)*cos(u)*log(u^2)*v*(1-v)/2, ((u^6)^(1/20)*(cos(u)^2)^(1/4)-1/2)*v*(1-v), v^(0.5)],
   ....:                        (u,0.001,1), (v,0,1), plot_points=[70,70], color='red')
   sage: p2 = parametric_plot3d([-sin(u)*cos(u)*log(u^2)*v*(1-v)/2, ((u^6)^(1/20)*(cos(u)^2)^(1/4)-1/2)*v*(1-v), v^(0.5)],
   ....:                        (u, 0.001,1), (v,0,1), plot_points=[70,70], color='red')
   sage: show(p1+p2)

A Hyperhelicoidal:

   sage: u = var("u")
   sage: v = var("v")
   sage: f_x = (sinh(v)*cos(3*u)) / (1+cosh(u)*cosh(v))
   sage: f_y = (sinh(v)*sin(3*u)) / (1+cosh(u)*cosh(v))
   sage: f_z = (cosh(v)*sinh(u)) / (1+cosh(u)*cosh(v))
   sage: parametric_plot3d([f_x, f_y, f_z], (u,-pi,pi), (v,-pi,pi), plot_points=[50,50], frame=False, color="red")
   Graphics3d Object

A Helicoid (lines through a helix,
http://en.wikipedia.org/wiki/Helix):

   sage: u, v = var('u,v')
   sage: f_x = sinh(v) * sin(u)
   sage: f_y = -sinh(v) * cos(u)
   sage: f_z = 3 * u
   sage: parametric_plot3d([f_x, f_y, f_z], (u,-pi,pi), (v,-pi,pi), plot_points=[50,50], frame=False, color="red")
   Graphics3d Object

Kuen's surface
(http://virtualmathmuseum.org/Surface/kuen/kuen.html):

   sage: f_x = (2*(cos(u) + u*sin(u))*sin(v))/(1+ u^2*sin(v)^2)
   sage: f_y = (2*(sin(u) - u*cos(u))*sin(v))/(1+ u^2*sin(v)^2)
   sage: f_z = log(tan(1/2 *v)) + (2*cos(v))/(1+ u^2*sin(v)^2)
   sage: parametric_plot3d([f_x, f_y, f_z], (u,0,2*pi), (v,0.01,pi-0.01), plot_points=[50,50], frame=False, color="green")
   Graphics3d Object

A 5-pointed star:

   sage: G1 = (abs(cos(u/4))^0.5+abs(sin(u/4))^0.5)^(-1/0.3)
   sage: G2 = (abs(cos(5*v/4))^1.7+abs(sin(5*v/4))^1.7)^(-1/0.1)
   sage: f_x = cos(u) * cos(v) * G1 * G2
   sage: f_y = cos(u) * sin(v) * G1 * G2
   sage: f_z = sin(u) * G1
   sage: parametric_plot3d([f_x, f_y, f_z], (u,-pi/2,pi/2), (v,0,2*pi), plot_points=[50,50], frame=False, color="green")
   Graphics3d Object

A cool self-intersecting surface (Eppener surface?):

   sage: f_x = u - u^3/3 + u*v^2
   sage: f_y = v - v^3/3 + v*u^2
   sage: f_z = u^2 - v^2
   sage: parametric_plot3d([f_x, f_y, f_z], (u,-25,25), (v,-25,25), plot_points=[50,50], frame=False, color="green")
   Graphics3d Object

The breather surface
(http://en.wikipedia.org/wiki/Breather_surface):

   sage: K = sqrt(0.84)
   sage: G = (0.4*((K*cosh(0.4*u))^2 + (0.4*sin(K*v))^2))
   sage: f_x = (2*K*cosh(0.4*u)*(-(K*cos(v)*cos(K*v)) - sin(v)*sin(K*v)))/G
   sage: f_y = (2*K*cosh(0.4*u)*(-(K*sin(v)*cos(K*v)) + cos(v)*sin(K*v)))/G
   sage: f_z = -u + (2*0.84*cosh(0.4*u)*sinh(0.4*u))/G
   sage: parametric_plot3d([f_x, f_y, f_z], (u,-13.2,13.2), (v,-37.4,37.4), plot_points=[90,90], frame=False, color="green")
   Graphics3d Object

var('x,y,z,u,v')
parametric_plot3d([u^2/v,v^2/u,u],(v,-2,2),(u,-2,2))

(x, y, z, u, v)

3D rendering not yet implemented

Definim els polinomis que ens donaran l'ideal.

f1 = X*(V)-U^2
f2 = Y*U-V^2
f3 = Z-U
f4 = 1-T*U*V

Construim l'ideal i calculem la base de Gröbner

J = A.ideal([f1,f2,f3,f4])
G=J.groebner_basis()

G

[T*X*Y - 1, T*Y*Z^2 - V, T*Z^3 - X, U - Z, V^2 - Y*Z, V*X - Z^2, V*Z - X*Y, X^2*Y - Z^3]

Trobem la base de Gröbner del segon ideal d'eliminació (format pels polinomis "sense U,V")

G3 = [g for g in G if g.degree(U)==0 and g.degree(V)==0 and g.degree(T)==0]
G3

[X^2*Y - Z^3]

"Comprovem" que la parametrització i la varietat son la mateixa

implicit_plot3d(G3[0].subs(X=x,Y=y,Z=z),(x,-2,2),(y,-2,2),(z,-2,2))

3D rendering not yet implemented

Per tal de poder veure que la parametrització abasta tots els punts de la varietat hem d'aplicar el teorema d'extensió dues vegades.

En el primer pas, donat un punt $(x,y,z)$ que és de la varietat parametritzada (és a dir, anul·la els polinomis del segon ideal d'eliminació) hem de veure que existeix un punt $(v,x,y,z)$ que és solució del primer ideal d'eliminació.

G1 = [g for g in G if g.degree(U)==0]
G1

[V^2 - Z, X^2 - Y*Z]

Per a cada polinomi de $G_1$, mirem quin és el seu terme líder, que serà de la forma $h_i(X,Y,Z)V^{N_i}$

Definim una funció auxiliar per calcular aquest polinomi $h_i$

def lcvar(f,X):
    """
    returns the leading coefficient of f viewed as a polynomial in the variable X
    """
    return f.coefficient(X^f.degree(X))

Calculem aquests polinomis per a tots els polinomis de $G_1$ (llevat dels de $G_2$, que no tenen $V$)

for g in G:
    if g.degree(T)>0:
        print "pol: %s\nlt:  %s\n" % (g,lcvar(g,T))

pol: T^2*Y - 3*T + X lt: Y pol: T*X - Y lt: X pol: T*Y^2 + X^2 - 3*Y lt: Y^2

Podrem aplicar el teorema d'extensió si el punt $(x,y,z)$ no és zero simultàniament de tots els polinomis $h_i(X,Y,Z)$.

En el nostre cas, com un dels polinomis $h_i$ és $1$ (en el primer cas, per exemple) aquesta condició es complirà sempre i, per tant, tota solució $(x,y,z)$ aixeca a una solució $(v,x,y,z)$

Tornem a aplicar el teorema d'extensió; ara per veure quines solucions $(v,x,y,z)$ del primer ideal d'eliminació aixequen a una solució global $(u,v,x,y,z)$

for g in G:
    if g.degree(U)>0:
        print "pol: %s\nlt:  %s\n" % (g,lcvar(g,U))

pol: U^2 - Y lt: 1 pol: U*V - X lt: V pol: U*X - V*Y lt: X pol: U*Z - V*X lt: Z

Pel mateix raonament que el cas anterior, veiem que tota solució parcial aixeca a una solució global. En resum: per a tot punt $(x,y,z)$ de la varietat parametritzada existeixen elements $u,v$ de manera que $(u,v,x,y,z)$ és solució de l'ideal $I$ (més tècnicament, un punt de la varietat $\mathbb{V}(I))$.

Dit d'altra manera: per a tot punt $(x,y,z)$ de la varietat paramatritzada existeix una elecció $(u,v)$ dels paràmetres de manera que $(x,y,z)=(f_1(u,v),f_2(u,v),f_3(u,v))$, és a dir, la parametrització abasta tots els punts.

Atenció!

El que hem fet funciona sobre un cos algebraicament tancat... Què passaria sobre un cos no algebraicament tancat?

En general, és difícil donar resposta a aquesta pregunta. En el cas que ens ocupa, podem veure que, per exemple, agafant com a cos base els nombres reals, la parametrització abasta tots els punts. En el primer pas, on a partir de $(x,y,z)$ s'ha d'extendre a una solució $(v,x,y,z)$ podem fer servir que el primer polinomi de $G_1$ és cúbic en $V$ i, per tant, sigui quin sigui $(x,y,z)$ tendrà al menys una arrel real. Per al segon pas, un argument "artesanal" per casos prova també que la solució local $(v,x,y,z)$ es pot aixecar a una solució global $(u,v,x,y,z)$

Segon exemple

Volem estudiar la varietat donada per la parametrització $$\begin{align} X&=UV\\ Y&=UV^2\\ Z&=U^2 \end{align}$$

A.<U,V,X,Y,Z>=PolynomialRing(QQ, order='lex')

f1 = X-U*V
f2 = Y-U*V^2
f3 = Z-U^2

Calculem la base de Gröbner de l'ideal i dels ideals d'eliminació.

J = A.ideal([f1,f2,f3])

G = J.groebner_basis()
G

[U^2 - Z, U*V - X, U*X - V*Z, U*Y - X^2, V^2*Z - X^2, V*X - Y, V*Y*Z - X^3, X^4 - Y^2*Z]

G1 = [g for g in G if g.degree(U)==0]
G2 = [g for g in G if g.degree(U)==0 and g.degree(V)==0]

G1

[V^2*Z - X^2, V*X - Y, V*Y*Z - X^3, X^4 - Y^2*Z]

G2

[X^4 - Y^2*Z]

La varietat parametritzada serà $$V=\mathbb{V}(X^4-Y^2Z)$$

Anem a aplicar el teorema d'extensió pas a pas.

for g in G1:
    if g.degree(V)>0:
        print "pol: %s\nlt:  %s\n" % (g,lcvar(g,V))

pol: V^2*Z - X^2 lt: Z pol: V*X - Y lt: X pol: V*Y*Z - X^3 lt: Y*Z

Els punts on no es pot aplicar el teorema d'extensió són aquells $(x,y,z)$ tals que $z=x=yz=0$, és a dir, aquells de la forma $(0,y,0)$ per a $y$ qualsevol, que observem que són de la varietat.

Per als altres punts, el teorema d'extensió assegura l'existència de $v$ tal que $(v,x,y,z)$ és solució.

Els punts $(0,y,0)$ amb $y\neq0$ no es poden abastar; en efecte, si avaluem els polinomis de $G_1$ en aquest punt obtenim:

[g.subs(X=0,Y=y,Z=0) for g in G1]

[0, -y, 0, 0]

Que no té solució ja que hem suposat que $y\neq0$.

Pel que respecta al segon pas:

for g in G:
    if g.degree(U)>0:
        print "pol: %s\nlt:  %s\n" % (g,lcvar(g,U))

pol: U^2 - Z lt: 1 pol: U*V - X lt: V pol: U*X - V*Z lt: X pol: U*Y - X^2 lt: Y

Com que un dels polinomis és mònic (vist com a polinomi en la variable $U$), tota solució parcial $(v,x,y,z)$ aixeca a una solució global $(u,v,x,y,z)$

En resum, la parametrització abasta tots els punts de la varietat, llevat dels de la forma $(0,y,0)$, amb $y\neq0$, que són de la varietat però no de la parametrització.

Vegem gràficament el que hem calculat:

g1=implicit_plot3d(x^4 - y^2*z,(x,-1,1),(y,-1,1),(z,-0.1,1),color='red')

g2=parametric_plot3d((u*v,u*v^2,u^2),(u,-1,1),(v,-1,1))

show(g1+g2)

3D rendering not yet implemented

Pregunta:

Analitzeu quins punts de la varietat s'abasten per la parametrització quan considerem $\mathbb{R}$ com a cos base.