Implicit Surfaces

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

# Cubic Surfaces

 Ding-Dong $x^2+y^2+z^2(z-1)=0$
 Cayley $-5(x^2(y+z)+y^2(x+z)+z^2(x+y))+2(xy+yx+xz)=0$
 Clebsch $81(x^3+y^3+z^3)-189(x^2(y+z)+y^2(x+z)+z^2(x+y))$$+54xyz+126(xy+yx+xz)-9(x^2+y^2+z^2)-9(x+y+z)+1=0  Nodal y^3+z^3-6yz=0  Witch of Agnesi a(y - 1) + (x^2 + z^2)y=0 a = 0.04 # Quartic Surfaces  Goursat - Rounded x^4+y^4+z^4+a(x^2+y^2+z^2)^2+b(x^2+y^2+z^2)+c=0 a = 0, b = 0, c = -1  Goursat - Tooth x^4+y^4+z^4+a(x^2+y^2+z^2)^2+b(x^2+y^2+z^2)+c=0 a = 0, b = -1, c = 0  Goursat - Tangle x^4+y^4+z^4+a(x^2+y^2+z^2)^2+b(x^2+y^2+z^2)+c=0 a = 0, b = -5, c = 11.8  Torricelli's trumpet (x^2+y^2)z^2-1=0  Cassini ((x-a)^2+y^2)((x-a)^2+y^2)-z^4=0 a = 0.5  Gerono lemniscate x^4-x^2+y^2+z^2=0  Chair (x^2+y^2+z^2-ak^2)^2-b((z-k)^2-2x^2)((z+k)^2-2y^2)=0 k = 5, a = 0.95, b = 0.8  Kummer x^4+y^4+z^4-x^2-y^2-z^2-x^2y^2-x^2z^2-y^2z^2+1=0  Roman x^2y^2+y^2z^2+x^2z^2+xyz=0  Diabolo x^2-(y^2+z^2)^2=0  Devil x^4+2(xz)^2-(0.6x)^2-y^4+(0.5y)^2+z^4=0  Gumdrop 4(x^4+(y^2+z^2)^2)+17x^2(y^2+z^2)-20(x^2+y^2+z^2)+17=0  Mitre 4x^2(x^2 + y^2 + z^2) - y^2(1 - y^2 - z^2)=0  Piriform (x-1)x^3 +y^2+z^2=0 # Quintic Surfaces  Kiss x^2+y^2+z^4(z-1)=0 # Sextic Surfaces  Heart (2x^2+y^2+z^2-1)^3-(1/10)x^2z^3-y^2z^3 = 0  Hunt 4(x^2+y^2+z^2-13)^3 + 27(3x^2+y^2-4z^2-12)^2 = 0  Barth 4(\phi^2x^2-y^2)(\phi^2y^2-z^2)(\phi^2z^2-x^2)-(1+2\phi)(x^2+y^2+z^2-1)^2=0 where \phi=(1+\sqrt{5})/2  Zitrus a^4(x^2 + z^2) + (y - 1)^3 y^3 = 0 a = 1  Kussner-Scmitt (x^2+3)(y^2+3)(z^2+3)-32(xyz+1)=0  Klein Bottle (x^2+y^2+z^2+2y-1)((x^2+y^2+z^2-2y-1)^2-8z^2)$$+16xz(x^2+y^2+z^2-2y-1)=0$
 McMullen K3 Model $(1 + x^2)(1 + y^2)(1 + z^2) + 8xyz - 2 = 0$
 Pilz $[ ( (x^2 + y^2 - 1)^2 + (z - 1)^2 ][ x^2/a^2 + (z - b)^2 - 1)^2 + y^2]-0.1= 0$ a = 1.4, b = 0.3

# Higher Order Algebraic Surfaces

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

# Non Algebraic Surfaces

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

# Chmutov Surfaces

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

# 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.