Generating surfaces from curvature – a playful approach

It is a common geometrical problem to design a motorway such that the steering wheel can be turned smoothly and uniformly (and not jumpy) while driving. Let us translate that into geometric terms. We are looking for a geometrical form of a curve (and therewith the curve itself) such that the curvature of that curve is as smooth and uniform as wished for. That is, we are looking for a curve for prescribed curvature data. Another family of examples of prescribed curvature are the minimal surfaces which can be modeled physically. They assume the shape of soap films spanned within curved wire frames. Mathematically, these minimal surfaces are characterized by vanishing mean curvature. If the mean curvature is constant but not vanishing we obtain so called constant mean curvature surfaces which appear as soap films too but with different air pressures on both sides of the film. For example as soap bubbles which have the form of a sphere or as soap bubbles which are attached to other objects like a wall and hence have not the shape of spheres. A very similar problem appears as the one of our project. Let us prescribe an arbitrary mathematical function (like sine, cosine, polynomial function or some arbitrary combination of well known functions). Then we determine a surface which assumes precisely that function as mean curvature. We choose for our approach the so called citizen science approach. For that we simplified some part of a larger project in such a way that it can be worked on by non-specialists. We expect a vast amount of experiments as a result of curiosity and willingness by the citizens. The resulting surfaces can be interesting from a mathematical point of view but also from an aesthetics point of view from which we will gain further insights. In particular we would like to test the method (surfaces from curvature) to its potential use as a design tool. One part of the project involves manufacturing models of the surfaces that we obtained in the first step (by experimentation and design). We manufacture the models by 3D printing. For that reason the 3D printing technology and all its related topics are an important part of the project. The involved citizen scientists will gain many important insights in this promising technology. The main target group of citizen scientists are pupils who take geometry classes. We expect that the chosen playful approach is very well suited for our target group who will like to generate many interesting and aesthetically pleasing examples of surfaces.

It is a common geometrical problem to design a motorway such that the steering wheel can be turned "smoothly" and uniformly (and not jumpy) while driving. Let us translate that into geometric terms. We are looking for a geometrical form of a curve (and therewith the curve itself) such that the curvature of that curve is as smooth and uniform as wished for. That is, we are looking for a curve for prescribed curvature data. Another family of examples of prescribed curvature are the minimal surfaces which can be modeled physically. They assume the shape of soap films spanned within curved wire frames. Mathematically, these minimal surfaces are characterized by vanishing mean curvature. If the mean curvature is constant but not vanishing we obtain so called constant mean curvature surfaces which appear as soap films too but with different air pressures on both sides of the film. For example as soap bubbles which have the form of a sphere or as soap bubbles which are attached to other objects like a wall and hence have not the shape of spheres. A very similar problem appears as the one of our project. Let us prescribe an arbitrary mathematical function (like sine, cosine, polynomial function or some arbitrary combination of well known functions). Then we determine a surface which assumes precisely that function as mean curvature. We choose for our approach the so called citizen science approach. For that we simplified some part of a larger project in such a way that it can be worked on by non-specialists. We expect a vast amount of experiments as a result of curiosity and willingness by the citizens. The resulting surfaces can be interesting from a mathematical point of view but also from an aesthetics point of view from which we will gain further insights. In particular we would like to test the method (surfaces from curvature) to its potential use as a design tool. One part of the project involves manufacturing models of the surfaces that we obtained in the first step (by experimentation and design). We manufacture the models by 3D printing. For that reason the 3D printing technology and all its related topics are an important part of the project. The involved citizen scientists will gain many important insights in this promising technology. The main target group of citizen scientists are pupils who take geometry classes. We expect that the chosen playful approach is very well suited for our target group who will like to generate many interesting and aesthetically pleasing examples of surfaces.