Implicit Surfaces

Quadric Surfaces

Sphere
$x^2+y^2+z^2-R^2=0$
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R = 1
Intersecting planes
$xy=0$
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Cubic Surfaces

Ding-Dong
$x^2+y^2+z^2(z-1)=0$
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Cayley
$-5(x^2(y+z)+y^2(x+z)+z^2(x+y))+2(xy+yx+xz)=0$
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Clebsch
$81(x^3+y^3+z^3)-189(x^2(y+z)+y^2(x+z)+z^2(x+y))$$+54xyz+126(xy+yx+xz)-9(x^2+y^2+z^2)-9(x+y+z)+1=0$
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Nodal
$y^3+z^3-6yz=0$
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Witch of Agnesi
$a(y - 1) + (x^2 + z^2)y=0$
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a = 0.04

Quartic Surfaces

Goursat - Rounded
$x^4+y^4+z^4+a(x^2+y^2+z^2)^2+b(x^2+y^2+z^2)+c=0$
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a = 0, b = 0, c = -1
Goursat - Tooth
$x^4+y^4+z^4+a(x^2+y^2+z^2)^2+b(x^2+y^2+z^2)+c=0$
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a = 0, b = -1, c = 0
Goursat - Tangle
$x^4+y^4+z^4+a(x^2+y^2+z^2)^2+b(x^2+y^2+z^2)+c=0$
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a = 0, b = -5, c = 11.8
Torricelli's trumpet
$(x^2+y^2)z^2-1=0$
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Cassini
$((x-a)^2+y^2)((x-a)^2+y^2)-z^4=0$
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a = 0.5
Gerono lemniscate
$x^4-x^2+y^2+z^2=0$
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Chair
$(x^2+y^2+z^2-ak^2)^2-b((z-k)^2-2x^2)((z+k)^2-2y^2)=0$
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k = 5, a = 0.95, b = 0.8
Kummer
$x^4+y^4+z^4-x^2-y^2-z^2-x^2y^2-x^2z^2-y^2z^2+1=0$
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Roman
$x^2y^2+y^2z^2+x^2z^2+xyz=0$
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Diabolo
$x^2-(y^2+z^2)^2=0$
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Devil
$x^4+2(xz)^2-(0.6x)^2-y^4+(0.5y)^2+z^4=0$
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Gumdrop
$4(x^4+(y^2+z^2)^2)+17x^2(y^2+z^2)-20(x^2+y^2+z^2)+17=0$
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Mitre
$4x^2(x^2 + y^2 + z^2) - y^2(1 - y^2 - z^2)=0$
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Piriform
$(x-1)x^3 +y^2+z^2=0$
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Quintic Surfaces

Kiss
$x^2+y^2+z^4(z-1)=0$
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Sextic Surfaces

Heart
$(2x^2+y^2+z^2-1)^3-(1/10)x^2z^3-y^2z^3 = 0$
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Hunt
$4(x^2+y^2+z^2-13)^3 + 27(3x^2+y^2-4z^2-12)^2 = 0$
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Barth
$4(\phi^2x^2-y^2)(\phi^2y^2-z^2)(\phi^2z^2-x^2)-(1+2\phi)(x^2+y^2+z^2-1)^2=0$ where $\phi=(1+\sqrt{5})/2$
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Zitrus
$a^4(x^2 + z^2) + (y - 1)^3 y^3 = 0$
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a = 1
Kussner-Scmitt
$(x^2+3)(y^2+3)(z^2+3)-32(xyz+1)=0$
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Klein Bottle
$(x^2+y^2+z^2+2y-1)((x^2+y^2+z^2-2y-1)^2-8z^2)$$+16xz(x^2+y^2+z^2-2y-1)=0$
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McMullen K3 Model
$(1 + x^2)(1 + y^2)(1 + z^2) + 8xyz - 2 = 0$
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Pilz
$[ ( (x^2 + y^2 - 1)^2 + (z - 1)^2 ][ x^2/a^2 + (z - b)^2 - 1)^2 + y^2]-0.1= 0$
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a = 1.4, b = 0.3

Higher Order Algebraic Surfaces

Bretzel (2 holes)
$(x^2(1-x^2)-y^2)^2+z^2-0.01=0$
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Bretzel (4 holes)
$0.1 (x^2(1.21-x^2)^2(3.8-x^2)^3-10y^2)^2+60z^2-2=0$
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Decocube
$((x^2+y^2-0.8^2)^2+(z^2-1)^2)((x^2+z^2-0.8^2)^2$$+(y^2-1)^2)((z^2+y^2-0.8^2)^2+(x^2-1)^2)-0.01=0$
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Horned Cube
$-3x^8-3y^8-2z^8+5(x^2yz)^2+3(xy^2z)^2+1=0$
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Linked Tori
$g(10x,10y−2,10z,13)g(10z,10y+2,10x,13)+1000 = 0$ where $g(x,y,z,c)=(x^2 +y^2 +z^2 +c)^2-53(x^2 +y^2)$
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Orthocircle
$[(x^2 + y^2-1)^2 + z^2] [(y^2 + z^2 - 1)^2 + x^2] [(z^2 + x^2 - 1) 2 + y^2]$$- a^2 [1 + b (x^2 + y^2 + z^2)]=0$
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a = 0.075 b = 3
Pretzel
$[ ((x-1)^2 + y^2 - 1) ((x+1)^2 + y^2 - 1) ]^2 + z^2 -0.1=0$
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Pseudo Cube
$x^{500}+y^{500}+z^{500}-1=0$
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Non Algebraic Surfaces

Blob
$x^2+y^2+z^2+\sin(4x)+\sin(4y)+\sin(4z)-1=0$
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Borg
$\sin(xy)+\sin(yz)+\sin(xz)=0$
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Scherk
$\exp(2z) * \cos(2y)-\cos(2x)=0$
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Torus
$(b-\sqrt{(x^2+y^2)})^2+z^2-a^2=0$
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a = 0.2, b = 1
Wiffle Cube
$1 - [ a^2 (x^2 + y^2 + z^2) ]^{-6} - [ b^8 (x^8 + y^8 + z^8) ]^6=0$
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a = 1/2.3, b = 1/2
P1 Atomic Orbital
$abs(x * \exp(-0.5(x^2+y^2+z^2)) -0.01=0$
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Chmutov Surfaces

4th degree
$T_4(x)+T_4(y)+T_4(z)=0$ where $T_4(x)=8x^4-8x^2-1$
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5th degree
$T_5(x)+T_5(y)+T_5(z)+1=0$ where $T_5(x)=16x^5-20x^3+5x$
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6th degree
$T_6(x)+T_6(y)+T_6(z)=0$ where $T_6(x)=32x^6-48x^4+18x^2-1$
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7th degree
$T_7(x)+T_7(y)+T_7(z)+1=0$ where $T_7(x)=64x^7-112x 5 +56x^3-7x$
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8th degree
$T_8(x)+T_8(y)+T_8(z)=0$ where $T_8(x)=128x^8-256x^6 + 160x^4-32x^2 + 1$
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9th degree
$T_9(x)+T_9(y)+T_9(z)+1=0$ where $T_9(x)=256x^9-576x^7 + 432x^5 - 120x^3 + 9x$
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Scene Files

The scene files for all surfaces on this page (plus some others) are available here in a Zip archive. Each surface is rendered from seven different viewpoints with three point light sources. The function plugins are also included in the archive.

Links

In spite of being already too large, this page is barely scratching the surface of the world of implicit shapes. For more fascinating shapes, look at the following places: