# Parametric Surface¶

Graphics 3D object for triangulating surfaces, and a base class for many other objects that can be represented by a 2D parametrization.

It takes great care to turn degenerate quadrilaterals into triangles and to propagate identified points to all attached polygons. This is not so much to save space as it is to assist the raytracers/other rendering systems to better understand the surface (and especially calculate correct surface normals).

AUTHORS:

• Robert Bradshaw (2007-08-26): initial version

EXAMPLES:

sage: from sage.plot.plot3d.parametric_surface import ParametricSurface, MoebiusStrip
sage: def f(x,y): return x+y, sin(x)*sin(y), x*y
sage: P = ParametricSurface(f, (srange(0,10,0.1), srange(-5,5.0,0.1)))
sage: show(P)

sage: S = MoebiusStrip(1,.2)
sage: S.is_enclosed()
False
sage: show(S)


By default, the surface is colored with one single color.

sage: P = ParametricSurface(f, (srange(0,10,0.1), srange(-5,5.0,0.1)),
....:  color="red")
sage: P.show()


One can instead provide a coloring function and a colormap:

sage: def f(x,y): return x+y, x-y, x*y
sage: def c(x,y): return sin((x+y)/2)**2
sage: cm = colormaps.RdYlGn
sage: P = ParametricSurface(f, (srange(-5,5,0.1), srange(-5,5.0,0.1)), color=(c,cm))
sage: P.show(viewer='tachyon')


Note that the coloring function should rather have values between 0 and 1. This value is passed to the chosen colormap.

Another colored example:

sage: colm = colormaps.autumn
sage: def g(x,y): return x, y, x**2 + y**2
sage: P = ParametricSurface(g, (srange(-10,10,0.1), srange(-5,5.0,0.1)), color=(c,colm))
sage: P.show(viewer='tachyon')


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).

Note

One may override eval() or eval_c() in a subclass rather than passing in a function for greater speed. One also would want to override get_grid.

Todo

actually remove unused points, fix the below code:

S = ParametricSurface(f=lambda xy: (xy[0],xy[1],0), domain=(range(10),range(10)))

class sage.plot.plot3d.parametric_surface.MoebiusStrip(r, width, twists=1, **kwds)

Base class for the MoebiusStrip graphics type. This sets the basic parameters of the object.

INPUT:

• r – a number which can be coerced to a float, serving roughly as the radius of the object

• width – a number which can be coerced to a float, which gives the width of the object

• twists – (default: 1) an integer, giving the number of twists in the object (where one twist is the ‘traditional’ Möbius strip)

EXAMPLES:

sage: from sage.plot.plot3d.parametric_surface import MoebiusStrip
sage: M = MoebiusStrip(3,3)
sage: M.show()

eval(u, v)

Return a tuple for $$x,y,z$$ coordinates for the given u and v for this MoebiusStrip instance.

EXAMPLES:

sage: from sage.plot.plot3d.parametric_surface import MoebiusStrip
sage: N = MoebiusStrip(7,3,2) # two twists
sage: N.eval(-1,0)
(4.0, 0.0, -0.0)

get_grid(ds)

Return appropriate $$u$$ and $$v$$ ranges for this MoebiusStrip instance.

This is intended for internal use in creating an actual plot.

INPUT:

• ds – A number, typically coming from a RenderParams object, which helps determine the increment for the $$v$$ range for the MoebiusStrip object.

EXAMPLES:

sage: from sage.plot.plot3d.parametric_surface import MoebiusStrip
sage: N = MoebiusStrip(7,3,2) # two twists
sage: N.get_grid(N.default_render_params().ds)
([-1, 1], [0.0, 0.12566370614359174, 0.25132741228718347, 0.37699111843077515, ...])

class sage.plot.plot3d.parametric_surface.ParametricSurface

Base class that initializes the ParametricSurface graphics type. This sets options, the function to be plotted, and the plotting array as attributes.

INPUT:

• f - (default: None) The defining function. Either a tuple of three functions, or a single function which returns a tuple, taking two python floats as input. To subclass, pass None for f and override eval_c or eval instead.

• domain - (default: None) A tuple of two lists, defining the grid of $$u,v$$ values. If None, this will be calculated automatically.

• color - (default: None) A pair $$(h,c)$$ where $$h$$ is a function with values in $$[0,1]$$ and $$c$$ is a colormap. The color of a point $$p$$ is then defined as the composition $$c(h(p))$$

EXAMPLES:

sage: from sage.plot.plot3d.parametric_surface import ParametricSurface
sage: def f(x,y): return cos(x)*sin(y), sin(x)*sin(y), cos(y)+log(tan(y/2))+0.2*x
sage: S = ParametricSurface(f, (srange(0,12.4,0.1), srange(0.1,2,0.1)))
sage: show(S)

sage: len(S.face_list())
2214


The Hessenberg surface:

sage: def f(u,v):
....:     a = 1
....:     from math import cos, sin, sinh, cosh
....:     x = cos(a)*(cos(u)*sinh(v)-cos(3*u)*sinh(3*v)/3) + sin(a)*(
....:         sin(u)*cosh(v)-sin(3*u)*cosh(3*v)/3)
....:     y = cos(a)*(sin(u)*sinh(v)+sin(3*u)*sinh(3*v)/3) + sin(a)*(
....:         -cos(u)*cosh(v)-cos(3*u)*cosh(3*v)/3)
....:     z = cos(a)*cos(2*u)*cosh(2*v)+sin(a)*sin(2*u)*sinh(2*v)
....:     return (x,y,z)
sage: v = srange(float(0),float((3/2)*pi),float(0.1))
sage: S = ParametricSurface(f, (srange(float(0),float(pi),float(0.1)),
....:                srange(float(-1),float(1),float(0.1))), color="blue")
sage: show(S)


A colored example using the color keyword:

sage: def g(x,y): return x, y, - x**2 + y**2
sage: def c(x,y): return sin((x-y/2)*y/4)**2
sage: cm = colormaps.gist_rainbow
sage: P = ParametricSurface(g, (srange(-10,10,0.1),
....:   srange(-5,5.0,0.1)),color=(c,cm))
sage: P.show(viewer='tachyon')

bounding_box()

Return the lower and upper corners of a 3D bounding box for self.

This is used for rendering and self should fit entirely within this box.

Specifically, the first point returned should have x, y, and z coordinates should be the respective infimum over all points in self, and the second point is the supremum.

EXAMPLES:

sage: from sage.plot.plot3d.parametric_surface import MoebiusStrip
sage: M = MoebiusStrip(7,3,2)
sage: M.bounding_box()
((-10.0, -7.53907349250478..., -2.9940801852848145), (10.0, 7.53907349250478..., 2.9940801852848145))

default_render_params()

Return an instance of RenderParams suitable for plotting this object.

dual()

Return an IndexFaceSet which is the dual of the ParametricSurface object as a triangulated surface.

EXAMPLES:

As one might expect, this gives an icosahedron:

sage: D = dodecahedron()
sage: D.dual()
Graphics3d Object


But any enclosed surface should work:

sage: from sage.plot.plot3d.shapes import Torus
sage: T =  Torus(1, .2)
sage: T.dual()
Graphics3d Object
sage: T.is_enclosed()
True


Surfaces which are not enclosed, though, should raise an exception:

sage: from sage.plot.plot3d.parametric_surface import MoebiusStrip
sage: M = MoebiusStrip(3,1)
sage: M.is_enclosed()
False
sage: M.dual()
Traceback (most recent call last):
...
NotImplementedError: this is only implemented for enclosed surfaces

eval(u, v)
get_grid(ds)
is_enclosed()

Return a boolean telling whether or not it is necessary to render the back sides of the polygons (assuming, of course, that they have the correct orientation).

This is calculated in by verifying the opposite edges of the rendered domain either line up or are pinched together.

EXAMPLES:

sage: from sage.plot.plot3d.shapes import Sphere
sage: Sphere(1).is_enclosed()
True

sage: from sage.plot.plot3d.parametric_surface import MoebiusStrip
sage: MoebiusStrip(1,0.2).is_enclosed()
False

jmol_repr(render_params)

Return a representation of the object suitable for plotting using Jmol.

json_repr(render_params)

Return a representation of the object in JSON format as a list with one element, which is a string of a dictionary listing vertices, faces and colors.

obj_repr(render_params)

Return a complete representation of object with name, texture, and lists of vertices, faces, and back-faces.

plot()

Draw a 3D plot of this graphics object, which just returns this object since this is already a 3D graphics object. Needed to support PLOT in doctrings, see trac ticket #17498

EXAMPLES:

sage: S = parametric_plot3d( (sin, cos, lambda u: u/10), (0, 20))
sage: S.plot() is S
True

tachyon_repr(render_params)

Return representation of the object suitable for plotting using Tachyon ray tracer.

threejs_repr(render_params)

Return a represention of the surface suitable for plotting with three.js.

EXAMPLES:

sage: _ = var('x,y')
sage: P = plot3d(x^2-y^2, (x, -2, 2), (y, -2, 2))
sage: P.threejs_repr(P.default_render_params())
[('surface',
{'color': '#6666ff',
'faces': [[0, 1, 2, 3],
...
'opacity': 1.0,
'vertices': [{'x': -2.0, 'y': -2.0, 'z': 0.0},
...
{'x': 2.0, 'y': 2.0, 'z': 0.0}]})]

triangulate(render_params=None)

Call self.eval_grid() for all $$(u,v)$$ in $$\text{urange} \times \text{vrange}$$ to construct this surface.

The most complicated part of this code is identifying shared vertices and shrinking trivial edges. This is not done so much to save memory, rather it is needed so normals of the triangles can be calculated correctly.

x3d_geometry()

Return XML-like representation of the coordinates of all points in a triangulation of the object along with an indexing of those points.