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I have just written a small program in C++ and OpenGl to plot implicit surfaces in $\mathbb{R}^3$ for a Graphical Computing class and now I'm in need of more interesting surfaces to implement!

Some that I've implemented are:

  • Basic surfaces like spheres and cylinders;
  • Nordstrand's Weird Surface;
  • Klein Quartic;
  • Goursat's Surface;
  • Heart Surface;

So, my question is, what are other interesting implicit surfaces in $\mathbb{R}^3?$

P.S.: I know this is kind of vague, but anything you find interesting will be of use. (:

P.P.S: Turn this into a community wiki, if need be.

t.b.
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Leonardo Fontoura
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    There is a wealth of documentation on Wolfram's Mathworld: http://mathworld.wolfram.com/topics/Surfaces.html – barf Jun 19 '11 at 01:38
  • Boy's surface.. – Cheerful Parsnip Jun 19 '11 at 01:45
  • If you type "mathematical surfaces" into Google you get a near endless list, including lots of raytraced models, for example: http://www.math.rug.nl/models/ – Ryan Budney Jun 19 '11 at 03:13
  • http://www.math.rug.nl/models/Clebsch.html is beautiful. Now I want one! Actually: can someone give me the equations of the lines in the Clebsh surface? – Mariano Suárez-Álvarez Jun 19 '11 at 03:24
  • @Mariano: I can't but [Oliver Lab](http://www.Cubics.AlgebraicSurface.net/data/diplomArbeit_OliverLabs.pdf) can (in German) see in particular chapter 3, but the general form looks rather messy. He has some cool pictures on his page [algebraicsurface.net](http://www.algebraicsurface.net/) – t.b. Jun 19 '11 at 15:40

4 Answers4

62

A really nice family of implicit surfaces in $\mathbb{R}^3$ are the Banchoff-Chmutov surfaces $$BC_n:=\{(x,y,z)\in\mathbb{R}^3\:;\:\: T_n(x)+T_n(y)+T_n(z)=0\},$$ where $T_n$ denotes the $n$-th Chebishev polynomial of the first kind, i.e. $$T_n(x):=\sum_{k=0}^{\lfloor n/2\rfloor} \binom{n}{2k} (x^2\!-\!1)^k x^{n-2k}.$$

  • a $\leftarrow BC_1\;\;\;$ $BC_2\rightarrow$ b
  • a $\leftarrow BC_3\;\;\;$ $BC_4\rightarrow$ b
  • a $\leftarrow BC_5\;\;\;$ $BC_6\rightarrow$ b
  • a $\leftarrow BC_7\;\;\;$ $BC_8\rightarrow$ b

What is the genus of $\mathbf{BC_{2n+2}}$? Code for plotting surfaces $BC_n$ in Mathematica 7:

BanchoffChmutov[n_] := ContourPlot3D[ChebyshevT[n,x]+ChebyshevT[n,y]+ChebyshevT[n,z], {x,-1.3,1.3}, {y,-1.3,1.3}, {z,-1.3,1.3}, Contours->0.02, AspectRatio->Automatic, Boxed->False, Axes->{False,False,False}, BoxRatios->Automatic, PlotRangePadding->None, PlotPoints->30, ViewPoint->{-2,3,3}]
surfacesBCn = Table[BanchoffChmutov[i], {i, 8}]
Table[ChebyshevT[n,x]+ChebyshevT[n,y]+ChebyshevT[n,z], {n, 8}]

Other miscellaneous increasingly pretty examples, that I managed to construct myself, include:

$$\begin{array}{r l} \small \{(x,y,z)\in\mathbb{R}^3\:;\:\: & \small (x-2)^2(x+2)^2+(y-2)^2(y+2)^2+(z-2)^2(z+2)^2+\\ & \small 3(x^2y^2+x^2z^2+y^2z^2)+6*x*y*z-10(x^2+y^2+z^2)+22=0\} \end{array}$$ enter image description here

$$\begin{array}{r l} \small \{(x,y,z)\in\mathbb{R}^3\:;\: & \small ((x-1)x^2(x+1)+y^2)^2+\\ & \small ((y-1)y^2(y+1)+z^2)^2+\\ & \small 0.1y^2+0.05(y-1)y^2(y+1)=0\} \end{array}$$ enter image description here

$$\begin{array}{r l} \small \{(x,y,z)\in\mathbb{R}^3\:;\: & \small 15((x-1.2)x^2(x+1.2)+y^2)^2+0.8(z-1)z^2(z+1)-0.1z^2+\\ & \small 20((y-0.8)y^2(y+0.8)+z^2)^2+0.8(x-1)z^2(x+1)-0.1x^2=0\} \end{array}$$ enter image description here

$$\begin{array}{r l} \small \{(x,y,z)\in\mathbb{R}^3\:;\: & \small ((x^2+y^2-0.85^2)^2+(z^2-1)^2)*\\ & \small ((y^2+z^2-0.85^2)^2+(x^2-1)^2)*\\ & \small ((z^2+x^2-0.85^2)^2+(y^2-1)^2)-0.001=0\} \end{array}$$ enter image description here

$$\begin{array}{r l} \small \{(x,y,z)\in\mathbb{R}^3\:;\: & \small (3(x-1)x^2(x+1)+2y^2)^2+(z^2-0.85)^2*\\ & \small (3(y-1)y^2(y+1)+2z^2)^2+(x^2-0.85)^2*\\ & \small (3(z-1)z^2(z+1)+2x^2)^2+(y^2-0.85)^2* -0.12=0\} \end{array}$$ enter image description here

$$\begin{array}{r l} \small \{(x,y,z)\in\mathbb{R}^3\:;\: & \small (2.92(x-1)x^2(x+1)+1.7y^2)^2*(y^2-0.88)^2+\\ & \small (2.92(y-1)y^2(y+1)+1.7z^2)^2*(z^2-0.88)^2+\\ & \small (2.92(z-1)z^2(z+1)+1.7x^2)^2*(x^2-0.88)^2 -0.02=0\} \end{array}$$ enter image description here

Hope you enjoy them...


Note also, that all of these examples have the form $\{(x,y,z)\in\mathbb{R}^3;\:P(x,y,z)=0\}=P^{-1}(0)$. They are $2$-manifolds (surfaces). To see this not only with your eyes but also in theory, compute the matrix $$\left[\frac{\partial P}{\partial x},\frac{\partial P}{\partial y},\frac{\partial P}{\partial z}\right]$$ at all the points of $P^{-1}(0)$ and find out (haven't done this myself) that the matrix is non-zero, which by implicit function theorem means that $P^{-1}(0)$ is a $3-1=2$ manifold.


Also, if you change the definition of $\{(x,y,z)\in\mathbb{R}^3;\:P(x,y,z)=0\}=P^{-1}(0)$ to $$\{(x,y,z)\in\mathbb{R}^3;\:P(x,y,z)\leq 0\}$$ or $$\{(x,y,z)\in\mathbb{R}^3;\:P(x,y,z)\geq 0\},$$ you get a $3$-manifold (because $0$ is a regular value of $P$) with the boundary $\{(x,y,z)\in\mathbb{R}^3;\:P(x,y,z)=0\}$, i.e. the surface $P^{-1}(0)$ with either it's interior or exterior "filled".


ADDITION (how to construct such surfaces): @Soarer, You never try to guess such a complicated polynomial. Here's the key to constructing such surfaces: step by step. You discover on the web, that the torus is $$\{P(x,y,z):=(x^2+y^2-0.7^2)^2+ z^2=0\}.$$ Then you learn that $$\{P(x-a,y-b,z-c)=0\}$$ is torus, translated by $(a,b,c)\in\mathbb{R}^3$. Also, you notice, that $$\{P(y,z,x)=0\}\text{ and }\{P(z,x,y)=0\}$$ is a torus with coordinate lines permuted (rotated by 90°) and $$\{P(ax,by,cz)=0\}$$ is a torus, stretched in x-direction by factor $1/a$, etc. Next, you see that $$\{P(x,y,z)\cdot Q(x,y,z)=0\}=\{P(x,y,z)=0\}\cup\{Q(x,y,z)=0\}.$$ Using all these techniques step by step, and being very patient, you manage to discover the following construction

enter image description here

using the code (Mathematica 7):

implicitSurface[p_String, zoom_, quality_Integer, view_List] := 
  ContourPlot3D[ToExpression[p], {x,-zoom,zoom}, {y,-zoom,zoom}, {z,-zoom,
  zoom}, Contours->0.02, Mesh->Automatic, AspectRatio->Automatic, Boxed->
  False, Axes->{False,False,False}, BoxRatios->Automatic, PlotRange->All, 
  PlotRangePadding->None, PlotPoints->quality, ViewPoint->view];
surfaces = {
  implicitSurface["(x^2+y^2-0.7^2)^2+ z^2", 1.2, 10, {0.3, -0.8, 1}],
  implicitSurface["(x^2+y^2-0.7^2)^2+(z^2-1)^2-0.05", 
                   1.2, 10, {1.3, -1.0, 2}],
  implicitSurface["(1.2y^2-1)^2*(x^2+y^2-1^2)^2+
                   (1.2z^2-1)^2*(y^2+z^2-1^2)^2+
                   (1.2x^2-1)^2*(z^2+x^2-1^2)^2-0.02", 
                   1.5, 10, {1.3, -1.0, 1.5}],
  implicitSurface["((x^2+y^2-0.85^2)^2+(z^2-1)^2)*
                   ((y^2+z^2-0.85^2)^2+(x^2-1)^2)*
                   ((z^2+x^2-0.85^2)^2+(y^2-1)^2)-0.001", 
                   1.2, 50, {1.2, -1.0, 1.3}],
  implicitSurface["(3(x-1)x^2(x+1)+y^2)^2+ 2z^2", 
                   1.2, 10, {0.3, -0.8, 1}],
  implicitSurface["(3(x-1)x^2(x+1)+y^2)^2+(2z^2-1)^2-0.005", 
                   1.2, 10, {0.3, -0.8, 1}],
  implicitSurface["(3(x-1)x^2(x+1)+2y^2)^2+(z^2-0.85)^2*
                   (3(y-1)y^2(y+1)+2z^2)^2+(x^2-0.85)^2*
                   (3(z-1)z^2(z+1)+2x^2)^2+(y^2-0.85)^2* -0.12", 
                   2, 80, {1, -1.8, 1.7}],
  implicitSurface["(2.92(x-1)x^2(x+1)+1.7y^2)^2*(y^2-0.88)^2+
                   (2.92(y-1)y^2(y+1)+1.7z^2)^2*(z^2-0.88)^2+
                   (2.92(z-1)z^2(z+1)+1.7x^2)^2*(x^2-0.88)^2 -0.02", 
                   2, 20, {0.4, -1.8, 1.3}]
  }

Of course, some of the examples were obtained just with a lot of trying and intelligent guessing.

It is apparent that one can construct surfaces, as complicated as one wishes, via these steps, but each time a new component is added ($P^{-1}(0) \mapsto (P\cdot Q)^{-1}(0)$), the degree of the polynomial increases, which presents considerable numerical problems when plotting the surface.


P.S. I hope I haven't made any mistakes when I copied the code. If so, please inform me, to check with my original Mathematica file.

P.P.S The above surfaces were included, as examples, in my diploma.

Leo
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Reposting my comment as requested:

There is a wealth of documentation on Wolfram's Mathworld: http://mathworld.wolfram.com/topics/Surfaces.html

I've also, not too long ago, seen a pdf file that had pictures and Mathematica code for a lot of these surfaces; if I can dig it out of my history I will post it as well.

barf
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I like this "tetrahedron":

enter image description here

$$ x^4 + 2x^2y^2 + 2x^2z^2 + y^4 + 2y^2z^2 + z^4 + 8xyz - 10x^2 - 10y^2 - 10z^2 + 20 = 0. $$

The Entzensberger star:

enter image description here

$$ 400(x^2y^2+ y^2z^2+ x^2z^2) = {(1-x^2-y^2-z^2)}^3. $$

On the higher space forum you can find amazing implicit surfaces, especially those of the member ICN5D. These are 3D slices of surfaces in higher dimension. I reproduced some of them with Python, like this one:

Solid Möbius strip, introduced in this paper:

Stéphane Laurent
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Comlpete minimal surfaces, discovered circa 1985. For example, Meeks' surfaces.

Andrew
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