Visualization (Pics)

Here some pictures of interesting surfaces, produced with jReality. For more images see

Minimal surfaces


In the research network "Minimal surfaces: integrable systems and visualisation" we explore integrable system methods for minimal, and related, surfaces. For visualisations of results, see


Darboux transforms of a homogeneous torus



Darboux transforms of a homogeneous torus which are tori of revolution  but not CMC in the 3-sphere [Le 2009]



Pseudo-Bubbleton jReality2 lobes (Darboux transform of the Clifford torus)



Compare to a Delaunay torus in the 3-sphere (torus of revolution in 3-sphere with constant mean curvature). Interesting coincidence?


Delaunay Torus













And a 3-lobed torus of revolution (not CMC in the 3-sphere).


3-lobes (Darboux transform of the Clifford torus)

3 lobes (Darboux transform of the Clifford torus), v2



More lobes.....



4 lobes (Darboux transform of Clifford torus)


5 lobes (Darboux transform of the Clifford torus)


CMC surfaces


Delaunay surface and Delaunay Bubbletons


DelaunayDelaunay Bubbleton Two step Bubbleton





Wente tori


Wente 43

Wente 43 open

Wente 65

Minimal surfaces

Helicoid and Catenoid





Willmore tori

Willmore cylinder and Willmore tori, all  obtained by a Darboux transform of a (covering of) the Clifford torus, [G-L 2009].



willmore cylinder clifford n-fold 1



clifford n-fold 2


















Hamiltonian Stationary Lagrangian tori


Castro Urbano tori [L-R 2008]



Castro Urbano 1Castro Urbano 2











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