Coverage for local/lib/python2.7/site-packages/sage/plot/plot3d/implicit_plot3d.py : 100%
Hot-keys on this page
r m x p toggle line displays
j k next/prev highlighted chunk
0 (zero) top of page
1 (one) first highlighted chunk
|
""" Implicit Plots """
r""" Plots an isosurface of a function.
INPUT:
- ``f`` - function
- ``xrange`` - a 2-tuple (x_min, x_max) or a 3-tuple (x, x_min, x_max)
- ``yrange`` - a 2-tuple (y_min, y_may) or a 3-tuple (y, y_min, y_may)
- ``zrange`` - a 2-tuple (z_min, z_maz) or a 3-tuple (z, z_min, z_maz)
- ``plot_points`` - (default: "automatic", which is 40) the number of function evaluations in each direction. (The number of cubes in the marching cubes algorithm will be one less than this). Can be a triple of integers, to specify a different resolution in each of x,y,z.
- ``contour`` - (default: 0) plot the isosurface f(x,y,z)==contour. Can be a list, in which case multiple contours are plotted.
- ``region`` - (default: None) If region is given, it must be a Python callable. Only segments of the surface where region(x,y,z) returns a number >0 will be included in the plot. (Note that returning a Python boolean is acceptable, since True == 1 and False == 0).
EXAMPLES::
sage: var('x,y,z') (x, y, z)
A simple sphere::
sage: implicit_plot3d(x^2+y^2+z^2==4, (x,-3,3), (y,-3,3), (z,-3,3)) Graphics3d Object
.. PLOT::
var('x,y,z') F = x**2 + y**2 + z**2 P = implicit_plot3d(F==4, (x,-3,3), (y,-3,3), (z,-3,3)) sphinx_plot(P)
A nested set of spheres with a hole cut out::
sage: implicit_plot3d((x^2 + y^2 + z^2), (x,-2,2), (y,-2,2), (z,-2,2), plot_points=60, contour=[1,3,5], ....: region=lambda x,y,z: x<=0.2 or y>=0.2 or z<=0.2, color='aquamarine').show(viewer='tachyon')
.. PLOT::
var('x,y,z') F = x**2 + y**2 + z**2 P = implicit_plot3d(F, (x,-2,2), (y,-2,2), (z,-2,2), plot_points=60, contour=[1,3,5], region=lambda x,y,z: x<=0.2 or y>=0.2 or z<=0.2, color='aquamarine') sphinx_plot(P)
A very pretty example, attributed to Douglas Summers-Stay (`archived page <http://web.archive.org/web/20080529033738/http://iat.ubalt.edu/summers/math/platsol.htm>`_)::
sage: T = RDF(golden_ratio) sage: F = 2 - (cos(x+T*y) + cos(x-T*y) + cos(y+T*z) + cos(y-T*z) + cos(z-T*x) + cos(z+T*x)) sage: r = 4.77 sage: implicit_plot3d(F, (x,-r,r), (y,-r,r), (z,-r,r), plot_points=40, color='darkkhaki').show(viewer='tachyon')
.. PLOT::
var('x,y,z') T = RDF(golden_ratio) F = 2 - (cos(x+T*y) + cos(x-T*y) + cos(y+T*z) + cos(y-T*z) + cos(z-T*x) + cos(z+T*x)) r = 4.77 V = implicit_plot3d(F, (x,-r,r), (y,-r,r), (z,-r,r), plot_points=40, color='darkkhaki') sphinx_plot(V)
As I write this (but probably not as you read it), it's almost Valentine's day, so let's try a heart (from http://mathworld.wolfram.com/HeartSurface.html)
::
sage: F = (x^2+9/4*y^2+z^2-1)^3 - x^2*z^3 - 9/(80)*y^2*z^3 sage: r = 1.5 sage: implicit_plot3d(F, (x,-r,r), (y,-r,r), (z,-r,r), plot_points=80, color='red', smooth=False).show(viewer='tachyon')
.. PLOT::
var('x,y,z') F = (x**2+9.0/4.0*y**2+z**2-1)**3 - x**2*z**3 - 9.0/(80)*y**2*z**3 r = 1.5 V = implicit_plot3d(F, (x,-r,r), (y,-r,r), (z,-r,r), plot_points=80, color='red', smooth=False) sphinx_plot(V)
The same examples also work with the default Jmol viewer; for example::
sage: T = RDF(golden_ratio) sage: F = 2 - (cos(x + T*y) + cos(x - T*y) + cos(y + T*z) + cos(y - T*z) + cos(z - T*x) + cos(z + T*x)) sage: r = 4.77 sage: implicit_plot3d(F, (x,-r,r), (y,-r,r), (z,-r,r), plot_points=40, color='deepskyblue').show()
Here we use smooth=True with a Tachyon graph::
sage: implicit_plot3d(x^2 + y^2 + z^2, (x,-2,2), (y,-2,2), (z,-2,2), contour=4, color='deepskyblue', smooth=True) Graphics3d Object
.. PLOT::
var('x,y,z') F = x**2 + y**2 + z**2 P = implicit_plot3d(F, (x,-2,2), (y,-2,2), (z,-2,2), contour=4, color='deepskyblue', smooth=True) sphinx_plot(P)
We explicitly specify a gradient function (in conjunction with smooth=True) and invert the normals::
sage: gx = lambda x, y, z: -(2*x + y^2 + z^2) sage: gy = lambda x, y, z: -(x^2 + 2*y + z^2) sage: gz = lambda x, y, z: -(x^2 + y^2 + 2*z) sage: implicit_plot3d(x^2+y^2+z^2, (x,-2,2), (y,-2,2), (z,-2,2), contour=4, ....: plot_points=40, smooth=True, gradient=(gx, gy, gz)).show(viewer='tachyon')
.. PLOT::
var('x,y,z') gx = lambda x, y, z: -(2*x + y**2 + z**2) gy = lambda x, y, z: -(x**2 + 2*y + z**2) gz = lambda x, y, z: -(x**2 + y**2 + 2*z) P = implicit_plot3d(x**2+y**2+z**2, (x,-2,2), (y,-2,2), (z,-2,2), contour=4, plot_points=40, smooth=True, gradient=(gx, gy, gz)) sphinx_plot(P)
A graph of two metaballs interacting with each other::
sage: def metaball(x0, y0, z0): return 1 / ((x-x0)^2+(y-y0)^2+(z-z0)^2) sage: implicit_plot3d(metaball(-0.6,0,0) + metaball(0.6,0,0), (x,-2,2), (y,-2,2), (z,-2,2), plot_points=60, contour=2, color='seagreen') Graphics3d Object
.. PLOT::
var('x,y,z') def metaball(x0, y0, z0): return 1 / ((x-x0)**2+(y-y0)**2+(z-z0)**2) P = implicit_plot3d(metaball(-0.6,0,0) + metaball(0.6,0,0), (x,-2,2), (y,-2,2), (z,-2,2), plot_points=60, contour=2, color='seagreen') sphinx_plot(P)
One can color the surface according to a coloring function and a colormap::
sage: t = (sin(3*z)**2).function(x,y,z) sage: cm = colormaps.gist_rainbow sage: G = implicit_plot3d(x^2 + y^2 + z^2, (x,-2,2), (y,-2,2), (z,-2, 2), ....: contour=4, color=(t,cm), plot_points=100) sage: G.show(viewer='tachyon')
.. PLOT::
var('x,y,z') t = (sin(3*z)**2).function(x,y,z) cm = colormaps.gist_rainbow G = implicit_plot3d(x**2 + y**2 + z**2, (x,-2,2), (y,-2,2), (z,-2, 2), contour=4, color=(t,cm), plot_points=60) sphinx_plot(G)
Here is another colored example::
sage: x, y, z = var('x,y,z') sage: t = (x).function(x,y,z) sage: cm = colormaps.PiYG sage: G = implicit_plot3d(x^4 + y^2 + z^2, (x,-2,2), ....: (y,-2,2),(z,-2,2), contour=4, color=(t,cm), plot_points=40) sage: G Graphics3d Object
.. PLOT::
x, y, z = var('x,y,z') t = (x).function(x,y,z) cm = colormaps.PiYG G = implicit_plot3d(x**4 + y**2 + z**2, (x,-2,2), (y,-2,2),(z,-2,2), contour=4, color=(t,cm), plot_points=40) sphinx_plot(G)
.. 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).
MANY MORE EXAMPLES:
A kind of saddle::
sage: implicit_plot3d(x^3 + y^2 - z^2, (x,-2,2), (y,-2,2), (z,-2,2), plot_points=60, contour=0, color='lightcoral') Graphics3d Object
.. PLOT::
x, y, z = var('x,y,z') G = implicit_plot3d(x**3 + y**2 - z**2, (x,-2,2), (y,-2,2), (z,-2,2), plot_points=60, contour=0, color='lightcoral') sphinx_plot(G)
A smooth surface with six radial openings::
sage: implicit_plot3d(-(cos(x) + cos(y) + cos(z)), (x,-4,4), (y,-4,4), (z,-4,4), color='orchid') Graphics3d Object
.. PLOT::
x, y, z = var('x,y,z') G = implicit_plot3d(-(cos(x) + cos(y) + cos(z)), (x,-4,4), (y,-4,4), (z,-4,4), color='orchid') sphinx_plot(G)
A cube composed of eight conjoined blobs::
sage: F = x^2 + y^2 + z^2 + cos(4*x) + cos(4*y) + cos(4*z) - 0.2 sage: implicit_plot3d(F, (x,-2,2), (y,-2,2), (z,-2,2), color='mediumspringgreen') Graphics3d Object
.. PLOT::
x, y, z = var('x,y,z') F = x**2 + y**2 + z**2 + cos(4*x) + cos(4*y) + cos(4*z) - 0.2 G = implicit_plot3d(F, (x,-2,2), (y,-2,2), (z,-2,2), color='mediumspringgreen') sphinx_plot(G)
A variation of the blob cube featuring heterogeneously sized blobs::
sage: F = x^2 + y^2 + z^2 + sin(4*x) + sin(4*y) + sin(4*z) - 1 sage: implicit_plot3d(F, (x,-2,2), (y,-2,2), (z,-2,2), color='lavenderblush') Graphics3d Object
.. PLOT::
x, y, z = var('x,y,z') F = x**2 + y**2 + z**2 + sin(4*x) + sin(4*y) + sin(4*z) - 1 G = implicit_plot3d(F, (x,-2,2), (y,-2,2), (z,-2,2), color='lavenderblush') sphinx_plot(G)
A Klein bottle::
sage: G = x^2 + y^2 + z^2 sage: F = (G+2*y-1)*((G-2*y-1)^2-8*z^2) + 16*x*z*(G-2*y-1) sage: implicit_plot3d(F, (x,-3,3), (y,-3.1,3.1), (z,-4,4), color='moccasin') Graphics3d Object
.. PLOT::
x, y, z = var('x,y,z') G = x**2 + y**2 + z**2 F = (G+2*y-1)*((G-2*y-1)**2-8*z**2)+16*x*z*(G-2*y-1) G = implicit_plot3d(F, (x,-3,3), (y,-3.1,3.1), (z,-4,4), color='moccasin') sphinx_plot(G)
A lemniscate::
sage: F = 4*x^2*(x^2+y^2+z^2+z) + y^2*(y^2+z^2-1) sage: implicit_plot3d(F, (x,-0.5,0.5), (y,-1,1), (z,-1,1), color='deeppink') Graphics3d Object
.. PLOT::
x, y, z = var('x,y,z') F = 4*x**2*(x**2+y**2+z**2+z) + y**2*(y**2+z**2-1) G = implicit_plot3d(F, (x,-0.5,0.5), (y,-1,1), (z,-1,1), color='deeppink') sphinx_plot(G)
Drope::
sage: implicit_plot3d(z - 4*x*exp(-x^2-y^2), (x,-2,2), (y,-2,2), (z,-1.7,1.7), color='darkcyan') Graphics3d Object
.. PLOT::
x, y, z = var('x,y,z') G = implicit_plot3d(z - 4*x*exp(-x**2-y**2), (x,-2,2), (y,-2,2), (z,-1.7,1.7), color='darkcyan') sphinx_plot(G)
A cube with a circular aperture on each face::
sage: F = ((1/2.3)^2 * (x^2 + y^2 + z^2))^(-6) + ((1/2)^8 * (x^8 + y^8 + z^8))^6 - 1 sage: implicit_plot3d(F, (x,-2,2), (y,-2,2), (z,-2,2), color='palevioletred') Graphics3d Object
.. PLOT::
x, y, z = var('x,y,z') F = ((1/2.3)**2 * (x**2 + y**2 + z**2))**(-6) + ((1/2)**8 * (x**8 + y**8 + z**8))**6 - 1 G = implicit_plot3d(F, (x,-2,2), (y,-2,2), (z,-2,2), color='palevioletred') sphinx_plot(G)
A simple hyperbolic surface::
sage: implicit_plot3d(x^2 + y - z^2, (x,-1,1), (y,-1,1), (z,-1,1), color='darkslategray') Graphics3d Object
.. PLOT::
x, y, z = var('x,y,z') G = implicit_plot3d(x**2 + y - z**2, (x,-1,1), (y,-1,1), (z,-1,1), color='darkslategray') sphinx_plot(G)
A hyperboloid::
sage: implicit_plot3d(x^2 + y^2 - z^2 -0.3, (x,-2,2), (y,-2,2), (z,-1.8,1.8), color='honeydew') Graphics3d Object
.. PLOT::
x, y, z = var('x,y,z') G = implicit_plot3d(x**2 + y**2 - z**2 -0.3, (x,-2,2), (y,-2,2), (z,-1.8,1.8), color='honeydew') sphinx_plot(G)
Dupin cyclide (:wikipedia:`Dupin_cyclide`) ::
sage: x, y, z , a, b, c, d = var('x,y,z,a,b,c,d') sage: a = 3.5 sage: b = 3 sage: c = sqrt(a^2 - b^2) sage: d = 2 sage: F = (x^2 + y^2 + z^2 + b^2 - d^2)^2 - 4*(a*x-c*d)^2 - 4*b^2*y^2 sage: implicit_plot3d(F, (x,-6,6), (y,-6,6), (z,-6,6), color='seashell') Graphics3d Object
.. PLOT::
x, y, z , a, b, c, d = var('x,y,z,a,b,c,d') a = 3.5 b = 3 c = sqrt(a**2 - b**2) d = 2 F = (x**2 + y**2 + z**2 + b**2 - d**2)**2 - 4*(a*x-c*d)**2 - 4*b**2*y**2 G = implicit_plot3d(F, (x,-6,6), (y,-6,6), (z,-6,6), color='seashell') sphinx_plot(G)
Sinus::
sage: implicit_plot3d(sin(pi*((x)^2+(y)^2))/2 + z, (x,-1,1), (y,-1,1), (z,-1,1), color='rosybrown') Graphics3d Object
.. PLOT::
x, y, z = var('x,y,z') G = implicit_plot3d(sin(pi*((x)**2+(y)**2))/2 + z, (x,-1,1), (y,-1,1), (z,-1,1), color='rosybrown') sphinx_plot(G)
A torus::
sage: implicit_plot3d((sqrt(x*x+y*y)-3)^2 + z*z - 1, (x,-4,4), (y,-4,4), (z,-1,1), color='indigo') Graphics3d Object
.. PLOT::
x, y, z = var('x,y,z') G = implicit_plot3d((sqrt(x*x+y*y)-3)**2 + z*z - 1, (x,-4,4), (y,-4,4), (z,-1,1), color='indigo') sphinx_plot(G)
An octahedron::
sage: implicit_plot3d(abs(x) + abs(y) + abs(z) - 1, (x,-1,1), (y,-1,1), (z,-1,1), color='olive') Graphics3d Object
.. PLOT::
x, y, z = var('x,y,z') G = implicit_plot3d(abs(x) + abs(y) + abs(z) - 1, (x,-1,1), (y,-1,1), (z,-1,1), color='olive') sphinx_plot(G)
A cube::
sage: implicit_plot3d(x^100 + y^100 + z^100 - 1, (x,-2,2), (y,-2,2), (z,-2,2), color='lightseagreen') Graphics3d Object
.. PLOT::
x, y, z = var('x,y,z') G = implicit_plot3d(x**100 + y**100 + z**100 - 1, (x,-2,2), (y,-2,2), (z,-2,2), color='lightseagreen') sphinx_plot(G)
Toupie::
sage: implicit_plot3d((sqrt(x*x+y*y)-3)^3 + z*z - 1, (x,-4,4), (y,-4,4), (z,-6,6), color='mintcream') Graphics3d Object
.. PLOT::
x, y, z = var('x,y,z') G = implicit_plot3d((sqrt(x*x+y*y)-3)**3 + z*z - 1, (x,-4,4), (y,-4,4), (z,-6,6), color='mintcream') sphinx_plot(G)
A cube with rounded edges::
sage: F = x^4 + y^4 + z^4 - (x^2 + y^2 + z^2) sage: implicit_plot3d(F, (x,-2,2), (y,-2,2), (z,-2,2), color='mediumvioletred') Graphics3d Object
.. PLOT::
x, y, z = var('x,y,z') F = x**4 + y**4 + z**4 - (x**2 + y**2 + z**2) G = implicit_plot3d(F, (x,-2,2), (y,-2,2), (z,-2,2), color='mediumvioletred') sphinx_plot(G)
Chmutov::
sage: F = x^4 + y^4 + z^4 - (x^2 + y^2 + z^2 - 0.3) sage: implicit_plot3d(F, (x,-1.5,1.5), (y,-1.5,1.5), (z,-1.5,1.5), color='lightskyblue') Graphics3d Object
.. PLOT::
x, y, z = var('x,y,z') F = x**4 + y**4 + z**4 - (x**2 + y**2 + z**2 - 0.3) G = implicit_plot3d(F, (x,-1.5,1.5), (y,-1.5,1.5), (z,-1.5,1.5), color='lightskyblue') sphinx_plot(G)
Further Chmutov::
sage: F = 2*(x^2*(3-4*x^2)^2+y^2*(3-4*y^2)^2+z^2*(3-4*z^2)^2) - 3 sage: implicit_plot3d(F, (x,-1.3,1.3), (y,-1.3,1.3), (z,-1.3,1.3), color='darksalmon') Graphics3d Object
.. PLOT::
x, y, z = var('x,y,z') F = 2*(x**2*(3-4*x**2)**2+y**2*(3-4*y**2)**2+z**2*(3-4*z**2)**2) - 3 G = implicit_plot3d(F, (x,-1.3,1.3), (y,-1.3,1.3), (z,-1.3,1.3), color='darksalmon') sphinx_plot(G)
Clebsch surface::
sage: F_1 = 81 * (x^3+y^3+z^3) sage: F_2 = 189 * (x^2*(y+z)+y^2*(x+z)+z^2*(x+y)) sage: F_3 = 54 * x * y * z sage: F_4 = 126 * (x*y+x*z+y*z) sage: F_5 = 9 * (x^2+y^2+z^2) sage: F_6 = 9 * (x+y+z) sage: F = F_1 - F_2 + F_3 + F_4 - F_5 + F_6 + 1 sage: implicit_plot3d(F, (x,-1,1), (y,-1,1), (z,-1,1), color='yellowgreen') Graphics3d Object
.. PLOT::
x, y, z = var('x,y,z') F_1 = 81 * (x**3+y**3+z**3) F_2 = 189 * (x**2*(y+z)+y**2*(x+z)+z**2*(x+y)) F_3 = 54 * x * y * z F_4 = 126 * (x*y+x*z+y*z) F_5 = 9 * (x**2+y**2+z**2) F_6 = 9 * (x+y+z) F = F_1 - F_2 + F_3 + F_4 - F_5 + F_6 + 1 G = implicit_plot3d(F, (x,-1,1), (y,-1,1), (z,-1,1), color='yellowgreen') sphinx_plot(G)
Looks like a water droplet::
sage: implicit_plot3d(x^2 +y^2 -(1-z)*z^2, (x,-1.5,1.5), (y,-1.5,1.5), (z,-1,1), color='bisque') Graphics3d Object
.. PLOT::
x, y, z = var('x,y,z') G = implicit_plot3d(x**2 +y**2 -(1-z)*z**2, (x,-1.5,1.5), (y,-1.5,1.5), (z,-1,1), color='bisque') sphinx_plot(G)
Sphere in a cage::
sage: F = (x^8+z^30+y^8-(x^4 + z^50 + y^4 -0.3)) * (x^2+y^2+z^2-0.5) sage: implicit_plot3d(F, (x,-1.2,1.2), (y,-1.3,1.3), (z,-1.5,1.5), color='firebrick') Graphics3d Object
.. PLOT::
x, y, z = var('x,y,z') F = (x**8+z**30+y**8-(x**4 + z**50 + y**4 -0.3)) * (x**2+y**2+z**2-0.5) G = implicit_plot3d(F, (x,-1.2,1.2), (y,-1.3,1.3), (z,-1.5,1.5), color='firebrick') sphinx_plot(G)
Ortho circle::
sage: F = ((x^2+y^2-1)^2+z^2) * ((y^2+z^2-1)^2+x^2) * ((z^2+x^2-1)^2+y^2)-0.075^2 * (1+3*(x^2+y^2+z^2)) sage: implicit_plot3d(F, (x,-1.5,1.5), (y,-1.5,1.5), (z,-1.5,1.5), color='lemonchiffon') Graphics3d Object
.. PLOT::
x, y, z = var('x,y,z') F = ((x**2+y**2-1)**2+z**2) * ((y**2+z**2-1)**2+x**2) * ((z**2+x**2-1)**2+y**2)-0.075**2 * (1+3*(x**2+y**2+z**2)) G = implicit_plot3d(F, (x,-1.5,1.5), (y,-1.5,1.5), (z,-1.5,1.5), color='lemonchiffon') sphinx_plot(G)
Cube sphere::
sage: F = 12 - ((1/2.3)^2 *(x^2 + y^2 + z^2))^-6 - ((1/2)^8 * (x^8 + y^8 + z^8))^6 sage: implicit_plot3d(F, (x,-2,2), (y,-2,2), (z,-2,2), color='rosybrown') Graphics3d Object
.. PLOT::
x, y, z = var('x,y,z') F = 12 - ((1/2.3)**2 *(x**2 + y**2 + z**2))**-6 - ( (1/2)**8 * (x**8 + y**8 + z**8) )**6 G = implicit_plot3d(F, (x,-2,2), (y,-2,2), (z,-2,2), color='rosybrown') sphinx_plot(G)
Two cylinders intersect to make a cross::
sage: implicit_plot3d((x^2+y^2-1) * (x^2+z^2-1) - 1, (x,-3,3), (y,-3,3), (z,-3,3), color='burlywood') Graphics3d Object
.. PLOT::
x, y, z = var('x,y,z') G = implicit_plot3d((x**2+y**2-1) * (x**2+z**2-1) - 1, (x,-3,3), (y,-3,3), (z,-3,3), color='burlywood') sphinx_plot(G)
Three cylinders intersect in a similar fashion::
sage: implicit_plot3d((x^2+y^2-1) * (x^2+z^2-1) * (y^2+z^2-1)-1, (x,-3,3), (y,-3,3), (z,-3,3), color='aqua') Graphics3d Object
.. PLOT::
x, y, z = var('x,y,z') G = implicit_plot3d((x**2+y**2-1) * (x**2+z**2-1) * (y**2+z**2-1)-1, (x,-3,3), (y,-3,3), (z,-3,3), color='aqua') sphinx_plot(G)
A sphere-ish object with twelve holes, four on each XYZ plane::
sage: implicit_plot3d(3*(cos(x)+cos(y)+cos(z)) + 4*cos(x)*cos(y)*cos(z), (x,-3,3), (y,-3,3), (z,-3,3), color='orangered') Graphics3d Object
.. PLOT::
x, y, z = var('x,y,z') G = implicit_plot3d(3*(cos(x)+cos(y)+cos(z)) + 4*cos(x)*cos(y)*cos(z), (x,-3,3), (y,-3,3), (z,-3,3), color='orangered') sphinx_plot(G)
A gyroid::
sage: implicit_plot3d(cos(x)*sin(y) + cos(y)*sin(z) + cos(z)*sin(x), (x,-4,4), (y,-4,4), (z,-4,4), color='sandybrown') Graphics3d Object
.. PLOT::
x, y, z = var('x,y,z') G = implicit_plot3d(cos(x)*sin(y) + cos(y)*sin(z) + cos(z)*sin(x), (x,-4,4), (y,-4,4), (z,-4,4), color='sandybrown') sphinx_plot(G)
Tetrahedra::
sage: implicit_plot3d((x^2+y^2+z^2)^2 + 8*x*y*z - 10*(x^2+y^2+z^2) + 25, (x,-4,4), (y,-4,4), (z,-4,4), color='plum') Graphics3d Object
.. PLOT::
x, y, z = var('x,y,z') G = implicit_plot3d((x**2+y**2+z**2)**2 + 8*x*y*z - 10*(x**2+y**2+z**2) + 25, (x,-4,4), (y,-4,4), (z,-4,4), color='plum') sphinx_plot(G)
TESTS:
Test a separate resolution in the X direction; this should look like a regular sphere::
sage: implicit_plot3d(x^2 + y^2 + z^2, (x,-2,2), (y,-2,2), (z,-2,2), plot_points=(10,40,40), contour=4) Graphics3d Object
Test using different plot ranges in the different directions; each of these should generate half of a sphere. Note that we need to use the ``aspect_ratio`` keyword to make it look right with the unequal plot ranges::
sage: implicit_plot3d(x^2 + y^2 + z^2, (x,0,2), (y,-2,2), (z,-2,2), contour=4, aspect_ratio=1) Graphics3d Object
sage: implicit_plot3d(x^2 + y^2 + z^2, (x,-2,2), (y,0,2), (z,-2,2), contour=4, aspect_ratio=1) Graphics3d Object
sage: implicit_plot3d(x^2 + y^2 + z^2, (x,-2,2), (y,-2,2), (z,0,2), contour=4, aspect_ratio=1) Graphics3d Object
Extra keyword arguments will be passed to show()::
sage: implicit_plot3d(x^2 + y^2 + z^2, (x,-2,2), (y,-2,2), (z,-2,2), contour=4, viewer='tachyon') Graphics3d Object
An implicit plot that doesn't include any surface in the view volume produces an empty plot::
sage: implicit_plot3d(x^2 + y^2 + z^2 - 5000, (x,-2,2), (y,-2,2), (z,-2,2), plot_points=6) Graphics3d Object
Make sure that implicit_plot3d doesn't error if the function cannot be symbolically differentiated::
sage: implicit_plot3d(max_symbolic(x, y^2) - z, (x,-2,2), (y,-2,2), (z,-2,2), plot_points=6) Graphics3d Object """ # These options, related to rendering with smooth shading, are irrelevant # since IndexFaceSet does not support surface normals: # smooth: (default: False) Whether to use vertex normals to produce a # smooth-looking surface. False is slightly faster. # gradient: (default: None) If smooth is True (the default), then # Tachyon rendering needs vertex normals. In that case, if gradient is None # (the default), then we try to differentiate the function to get the # gradient. If that fails, then we use central differencing on the scalar # field. But it's also possible to specify the gradient; this must be either # a single python callable that takes (x,y,z) and returns a tuple (dx,dy,dz) # or a tuple of three callables that each take (x,y,z) and return dx, dy, dz # respectively.
|