Several views of a Klein surface (often called Klein bottle): principal directions, mean and Gaussian curvature, transparent view, surface view with reflections


A real-time visualisation tool for implicit surfaces, i.e. isolevels of real-value functions f(x,y,z). It casts rays through each pixel and computes intersections with the surface.

Go to AlgoSurfer Demo | AlgoSurfer

Main features

  • it displays any implicit surface, either predefined or user-written, in real time.
  • it displays either the surface, its normals, its mean or Gaussian curvatures or its principal directions.
  • you can choose different illumination models (basic, Phong) but also use transparent materials or reflecting materials
  • you can navigate around the shape with the mouse

Predefined shapes

The tool offers several classical algebraic (Cayley Cubic, Whitney surface, etc) or implicit surfaces (Sierpinski cube, Kleinian variation, etc).

View of mean curvatures onto Kleinian variation
View of mean curvatures onto Kleinian variation

Writing your own implicit function

Just write it in the textbox like:

// Torus example
return (x2+y2+z2+a*a-b*b)*(x2+y2+z2+a*a-b*b)-4.*a*a*(x2+y2);

Obviously, you have access to x, y, z variables (and even a temporal t), as well as their squared and cubic variants (x2, x3, …). Use x2*x3 if you want x^5.

You can also access user-interface parameters a, b, c.

The code should be written in GLSL (C-like) and then press View Surface. If you do not know GLSL, the simplest way is to take inspiration of the code written for predefined shapes.


  • you can change the viewed isolevel (default is 0).
  • scale allows you to zoom in/out of the surface.
  • center_x, center_y, center_z changes the center of view
  • you may decrease accuracy to get better details, at the price of increased computational burden (check fps above)
  • singularity controls how singularity are detected, 1.0: large band 0.0: invisible.
  • you may change the color_model to change the color display of surfaces
    • Basic: just the natural color as defined by what you are viewing
    • Phong: Adds a simple Phong model with a front white light
    • Phong+lights: Adds a Phong model with two other parametrizable lights
  • you may change the colors of the surface (diffuse inside/outside), specular, singularities.
  • you may change lights colors and directions.
  • view_shape changes the volume that encloses the isosurface.
  • view_radius changes the sizes of the previous volume.
  • grid_size, grid_thickness and grid_attenuation changes the display of overlayed grids/spheres/isolines.
  • rotation_x, rotation_y, rotation_z changes the current object rotation speed.
  • stop stops the rotations.
Jacques-Olivier Lachaud
Jacques-Olivier Lachaud
Professor of Computer Science

My research interests include digital geometry, geometry processing, image analysis, variational models and discrete calculus.