# Implicit Plots¶

sage.plot.plot3d.implicit_plot3d.implicit_plot3d(f, xrange, yrange, zrange, **kwds)

Plot 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_max) or a 3-tuple (y, y_min, y_max)
• zrange – a 2-tuple (z_min, z_max) or a 3-tuple (z, z_min, z_max)
• 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


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')


A very pretty example, attributed to Douglas Summers-Stay (archived page):

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')


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')


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


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')


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


One can also color the surface using a coloring function and a colormap as follows. Note that the coloring function must take values in the interval [0,1].

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')


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


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:

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


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


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


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


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


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


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


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


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


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


Dupin cyclide (Wikipedia article 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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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