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# Visualization (Labs)

Here some interactive labs of minimal, CMC, HSL and Willmore surfaces

# Labs

Labs

To start a lab just click on the screenshot. If the application does not start, have a look at the Help page at TU Berlin.

This lab shows a 1-parameter family of Double Enneper surfaces (converging to the catenoid as the parameter goes to infinity).

These examples are computed by the methods in [LR].

Warning; this is still a rather incomplete applet.

Here is what you see: n is the maximal period after which the Darboux transforms close; k and l are free (integer) parameters, with k*k + l*l < 2*n*n.

Unfortunately, you still have to adjust the periods  by hand, that is if n=6 you have to have x max = y max =6. Note that there are examples with smaller periods.

The green point in the left plane is a free (complex) parameter, you can move it around.

This lab shows the μ-Darboux transforms (see CLP) of Delaunay surfaces. Uses heavily a Delaunay implementation and the elliptic functions library of Nick Schmitt, Universität Tübingen. Thanks also to Nick Schmitt and Alexander Gouberman for the refining programs.

 By setting the necksize, you can choose whether you want to see the μ-Darboux transforms of a cylinder (=0.5), an unduloid (>0) or a nodoid (<0). The lower left plane is the loop parameter μ of the family of flat connections coming with a constant mean curvature surface. The upper left plane parametrizes the set of all μ-Darboux transforms for a fixed μ by a complex m. Moving the latter parameter around you will see that, generically, the Darboux transform is not closed. Howevever, for m=0 the surface is always closed. At exceptional points (the branch points of the projection of the spectral curve to the complex plane), Bubbletons occur. These points are marked by the red points in the μ-plane. If you want to see Bubbletons move the μ-parameter onto the red points (or push the "go to bubbleton" button). You will see a bubbleton once you move the parameter in the upper left plane. The number of lobes can be changed by moving to another red point (or change the "number of lobes" and push again the "go to bubbleton" button). Note that you can change the grid to get more accurate 1-Bubbeltons. Moreover, you can adjust the refiner by asking for lower areas and/or higher number of refinement steps (max refining).

This lab shows Wente tori and is completely due to Nick Schmitts CMC library which gives the java classes for Wente tori. I just provided the oorange network...

 This lab is based on the Wente-java library of Nick Schmitt. By choosing the parameters L, N so that L/N is in the open interval from 1 to 2 one obtains Wente tori with more lobes.

This lab shows the μ-Darboux transforms (see CLP) of a Wente torus. Thanks to Nick Schmitt for the underlying CMC library.

 The upper left plane is the loop parameter μ of the family of flat connections coming with a cmc surface. The lower left plane parametrizes the set of all μ-Darboux transforms for a fixed μ. Moving the latter parameter m around in the plane you will see that generically the Darboux transform is not closed. Howevever, when you push the "reset m" button, that is for m=0, you will aways obtain a closed surface. We are still investigating if there are "bubbleton type" surfaces. Note that you can change the grid to get more accurate Wente tori. Moreover, you can adjust the refiner by asking for lower areas and/or higher number of refinement steps (max refining).

This lab shows the μ-Darboux transforms (see CLP) of a 1-Bubbleton. Thanks to Nick Schmitt and Alexander Gouberman for the refining programs.

 The lower left plane is the loop parameter μ of the family of flat connections coming with a cmc surface. The upper left plane parametrizes the set of all μ-Darboux transforms for a fixed μ. Moving the latter parameter around in the plane you will see that generically the Darboux transform is not closed. Howevever, when you push the "reset m" button you will aways obtain a closed surface. If you want to see 2-Bubbletons move the μ-parameter onto the red points (or push the "go to bubbleton" button). You will see a 2-bubbleton once you move the parameter in the upper left plane. The number of lobes can be changed by moving to another red point (or change the "number of lobes" and push again the "go to bubbleton" button. Additionally you have the option of moving the lobes on the original 1-Bubbleton (translational parameter) and to adjust the number of lobes of the original 1-Bubbleton. Note that you can change the grid to get more accurate 1- and 2-Bubbeltons. Moreover, you can adjust the refiner by asking for lower areas and/or higher number of refinement steps (max refining).

This lab shows the μ-Darboux transforms (see RL) for μ=-1 of the Clifford torus. The μ-Darboux transforms were computed by using the family of flat connections, which is given by the harmonic left normal of the Clifford torus.

 Free parameters are the green point in the upper plane which gives (open) Darboux transforms of the Clifford torus, and the yellow point in the lower plane. If the green point is the origin, then any value of μ gives a closed μ-Darboux transform of the Clifford torus. The red points in the applet indicate the μ's for which one period can be closed. Moving μ to the red point (or alternatively, chosing one of the "go to bubbleton" buttons, and adjust the integer to your fancy) you can see that either the x or y period closes, whereas the other period only gives in the limit x, y -> ∞ a closed surface.

This lab shows the associated family of the helicoid.

Use the spectral parameter to transform the helicoid into the catenoid.

 This lab shows a family of Castro Urbano tori, see [LR]. This family shows a deformation of a (3-fold covering of a) rectangular torus, all surfaces are Hamiltonian Stationary tori. The family is due to Castro/ Urbano.

Contact details

Department of Mathematics
University of Leicester
Leicester LE1 7RH
United Kingdom

Tel.: +44 (0)116 252 3917
Fax: +44 (0)116 252 3915

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