Superformula

 

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 The Superformula is a mathematical expression that enables 2D- and 3D geometry definition with one single expression. Johan Gielis discovered the basic Superformula from Gabriel Lamé in 1995 and used it to represent biological structures. In 1996, he and Bert Beirinckx started developing the Superformula with first results in 1997 producing multisymmetrical structures and a first 3D structure.

Throughout the years, the formula was further developed with 2D and 3D-applications and a book was published: “Inventing the Circle” that you can purchase via www.geniaal.be, the company Johan started in 1999. For more info on this book, you can also visit Amazon.com .

At the end of 2002, Genicap Corporation NV was born with emphasis on technological development of engineering and graphics solutions.

Core Technology News

  • 14/07/2004: [3D] Superformula enters the realm of differential geometry.
  • 13/07/2004: [General] The Superformula speaks its first words in audio: first results on SuperSounds
  • 06/07/2004: [2D] Major update on C-Points technology enabling better performance in calculation and shape generation
  • 26/06/2004: [General] Contour plots make some interesting wallpapers
  • 26/11/2003: [3D] 1st Phase development of generalised cylinders finalised
  • 06/11/2003: [3D] Theoretical development on Superformula Rings finalised
  • 30/10/2003: [3D] First version of the 3D modeler creates its first models in incredibly small filesizes, results astonish even the creators of the Superformula
  • 30/09/2003: [General] Small applications developed like a wing design tool (including asymmetric wings and lift calculation), shape calculation tool and more
  • 24/05/2003: [3D] Generalised Calculations of 3D objects is a fact, Optimisation calculations are now possible in 3D
  • 15/04/2003: [3D] 3D parametric formula passes tests and is to be implemented in 3D modeler
  • 12/02/2003: [2D] Generalised Calculation of shape properties (area, curvature, circumference etc etc) is a fact, shape optimisation a real opportunity
  • 14/01/2003: [2D] Research on creation of straight-line polygons finished
  • 02/12/2002: [General] Start of Genicap Corporation NV, Bert is first employee and re-ignites development of the Superformula technology for engineering applications
  • ??/??/1999: Johan starts Geniaal BVBA
  • 1997-2002: Bert starts his career as an engineer while development of the Supershape technology is taking the slow lane, but not for too long...
  • ??/07/1997: Johan breaks the code and develops multisymmetric formula
  • ??/05/1997: Bert defends his thesis succesfully: Lamé-ovals are translated into polar coördinates, a crude 3D technology is generated and first properties in 2D like surface and moments of inertia are developed for elementary shapes
  • ??/12/1996: Symposium of morphogenesis of Plants, first ideas solidify
  • 08/05/1996: Bert Beirinckx and Johan Gielis meet the first time for Bert's thesis at KdG Hogeschool campus Hoboken, the start of The Superformula Era

Sneak Preview to future Projects

If you think you saw it all... think again...

The Superformula does not only apply on shapes or graphics. It offers a whole new means of designing, engineering and optimizing concepts, products and processes. But first things first...

Graphics

Geometry definition goes much further than just blocks, closed shapes and tori. The next step will be generalised cilinders. The concept is ready and operational, software development is ongoing. The pictures below will give some idea on possibilities with generalised cilinders...

This is a cilinder that is swerving over a 3D wave: cos(t) in XZ-plane and sin(t) in YZ-plane. Along the path, the shape transforms from a circle to a starfish. Of course, any shape is feasable as the same geometric formula is used as in all other Genicap software. The path itself can also be controlled by the Superformula.

Unified Calculations

Next up are unified calculations. Based on the Superformula, generic techniques have been developed to calculate circumferences, surfaces, volumes, mass moments and all needed geometric values of geometries. This means that only one generic formula is needed for every property needed, the same parameters that are used in the geometric Superfomula will calculate the desired property.

This is very interesting in CAE-software, where a very big simplification of programming can be accomplished.  In OOP you only need one small piece of code to calculate the property of all kinds of shapes.

Optimization

Thanks to the unified calculation techniques, all sorts of engineering applications are feasable. Ever questioned which kind of geometry suits your application best? Let's take a very simple example: yoghurt pots.

This is a product that is produced in millions a day, so what if we could save on material, storage and transportation costs? Therefore we would like to have a pot that contains its yoghurt in the cheapest possible way, the most compact way. To make the problem more complex, we need to keep a fixed height, set by the customer.

Then we get to work, or better, let the Superformula do it. We simple model the design and tell the system which shapes it can alter considering size and geometry.

We will examine changes in the circular cross section, keeping the volume of the container as a fixed constraint and have the total amount of plastic displayed in a graphic.  We vary the changes in the cross sectional area by changing the Superformula parameter n (= n1 = n2 = n3).

Analytically, a parameter of 3.2 would give the optimal result, but what would it look like?

Have a look at the dimensions and compare them with the initial design. There are some interesting reductions in size and in weight as well. This design takes up 2% less material than the original one.

With this technique, a new, exact way of analysis is possible by having the geometry represented by a mere parameter, a number. You can have the exact analysis without the need for an expensive, parametric 3D design tool that has a huge cost on licensing, training and hardware.