# Implicit Plots¶

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

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 50) 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).show(viewer='tachyon')


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

sage: T = RDF(golden_ratio)
sage: p = 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(p, (x, -r, r), (y, -r, r), (z, -r, r), plot_points=40).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: p = (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(p, (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: p = 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(p, (x, -r, r), (y, -r, r), (z, -r, r), plot_points=40).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, 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)
Graphics3d Object


One can color the surface according to a coloring function and a colormap:

sage: t = (sin(2*y+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=60)
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)
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))
Graphics3d Object


A cube composed of eight conjoined blobs:

sage: implicit_plot3d(x^2 + y ^2 + z^2 +cos(4*x)+cos(4*y)+cos(4*z)-0.2, (x, -2, 2), (y, -2, 2), (z, -2, 2))
Graphics3d Object


A variation of the blob cube featuring heterogeneously sized blobs:

sage: implicit_plot3d(x^2 + y ^2 + z^2 +sin(4*x) + sin(4*y) + sin(4*z) -1, (x, -2, 2), (y, -2, 2), (z, -2, 2))
Graphics3d Object


A klein bottle:

sage: implicit_plot3d((x^2+y^2+z^2+2*y-1)*((x^2+y^2+z^2-2*y-1)^2-8*z^2)+16*x*z*(x^2+y^2+z^2-2*y-1), (x, -3, 3), (y, -3.1, 3.1), (z, -4, 4))
Graphics3d Object


A lemniscate:

sage: implicit_plot3d(4*x^2*(x^2+y^2+z^2+z)+y^2*(y^2+z^2-1), (x, -0.5, 0.5), (y, -1, 1), (z, -1, 1))
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))
Graphics3d Object


A cube with a circular aperture on each face:

sage: implicit_plot3d(((1/2.3)^2 *(x^2 + y^2 + z^2))^-6 + ( (1/2)^8 * (x^8 + y^8 + z^8) )^6 -1, (x, -2, 2), (y, -2, 2), (z, -2, 2))
Graphics3d Object


A simple hyperbolic surface:

sage: implicit_plot3d(x*x + y - z*z, (x, -1, 1), (y, -1, 1), (z, -1, 1))
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))
Graphics3d Object


Duplin cycloid:

sage: implicit_plot3d((2^2 - 0^2 - (2 + 2.1)^2) * (2^2 - 0^2 - (2 - 2.1)^2)*(x^4+y^4+z^4)+ 2*((2^2 - 0^2 - (2 + 2.1)^2 )*(2^2 - 0^2 - (2 - 2.1)^2)* (x^2 * y^2+x^2 * z^2+y^2 * z^2))+2* 2^2 *((-0^2-2^2+2^2+2.1^2)* (2 *x *2+2* y* 0-2^2)-4*0 *2.1^2 *y)*(x^2+y^2+z^2)+ 4 * 2^4 * (2 *x+0 *y)* (-2^2+0 * y+2 * x)+4* 2^4 * 2.1^2 * y^2+2^8, (x, -2, 2.2), (y, -2, 2), (z, -1.3, 1.3))
Graphics3d Object


Sinus:

sage: implicit_plot3d(sin(pi*((x)^2+(y)^2))/2 +z, (x, -1, 1), (y, -1, 1), (z, -1, 1))
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))
Graphics3d Object


An octahedron:

sage: implicit_plot3d(abs(x)+abs(y)+abs(z) - 1, (x, -1, 1), (y, -1, 1), (z, -1, 1))
Graphics3d Object


A cube:

sage: implicit_plot3d(x^100 + y^100 + z^100 -1, (x, -2, 2), (y, -2, 2), (z, -2, 2))
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))
Graphics3d Object


A cube with rounded edges:

sage: implicit_plot3d(x^4 + y^4 + z^4 - (x^2 + y^2 + z^2), (x, -2, 2), (y, -2, 2), (z, -2, 2))
Graphics3d Object


Chmutov:

sage: implicit_plot3d(x^4 + y^4 + z^4 - (x^2 + y^2 + z^2-0.3), (x, -1.5, 1.5), (y, -1.5, 1.5), (z, -1.5, 1.5))
Graphics3d Object


Further Chutmov:

sage: implicit_plot3d(2*(x^2*(3-4*x^2)^2+y^2*(3-4*y^2)^2+z^2*(3-4*z^2)^2) -3, (x, -1.3, 1.3), (y, -1.3, 1.3), (z, -1.3, 1.3))
Graphics3d Object


Clebsch:

sage: implicit_plot3d(81*(x^3+y^3+z^3)-189*(x^2*y+x^2*z+y^2*x+y^2*z+z^2*x+z^2*y) +54*x*y*z+126*(x*y+x*z+y*z)-9*(x^2+y^2+z^2)-9*(x+y+z)+1, (x, -1, 1), (y, -1, 1), (z, -1, 1))
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))
Graphics3d Object


Sphere in a cage:

sage: implicit_plot3d((x^8 + z^30 + y^8 - (x^4 + z^50 + y^4 -0.3))*(x^2 + y^2 + z^2 -0.5), (x, -1.2, 1.2), (y, -1.3, 1.3), (z, -1.5, 1.5))
Graphics3d Object


Ortho circle:

sage: implicit_plot3d(((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)), (x, -1.5, 1.5), (y, -1.5, 1.5), (z, -1.5, 1.5))
Graphics3d Object


Cube sphere:

sage: implicit_plot3d(12 - ((1/2.3)^2 *(x^2 + y^2 + z^2))^-6 - ( (1/2)^8 * (x^8 + y^8 + z^8) )^6, (x, -2, 2), (y, -2, 2), (z, -2, 2))
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))
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))
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))
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))
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))
Graphics3d Object


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


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