How to Expand and Contract Curved Surfaces of all Shapes

Making an object bigger or smaller is usually only possible by stretching it, crumpling it or changing its shape in some other way. Structures that can change their size without changing shape are called dilational. Such devices can have important applications in engineering and medicine – think stents placed in human arteries, for example. Current dilational mechanisms are limited to very few shapes, mostly spheres or sphere-like surfaces. A well-known example is the children’s toy based on Hoberman’s sphere, where the joints fold into the centre of the ball as it contracts. Such mechanisms have the disadvantage that the parts that enable the object to expand and contract, move at an angle, usually perpendicular to the surface of the object. This means that as the object changes shape, the mechanical parts either stick out or protrude into the enclosed volume. That is far from ideal for many applications; it would hamper the flow of blood in the case of aortic stents, for example.

Triangulation + pantograph = dilation

Freek Broeren and Werner van de Sande, researchers at TU Delft’s department of Precision and Microsystems Engineering (PME), have designed a dilation method that can be applied to any curved surface. For this, they used triangulation, the visualisation of a curved object by means of triangles placed all over the surface. Triangle meshes are an often-used, computation-efficient way to represent 3d structures in computer graphics. They combined this 21st century ingenuity with the 17th century pantograph: a device first attested in literature in 1653, made of four bars fixed at one point and pivoted at the others. It is used to scale up drawings, for example. Broeren and Van de Sande used the concept of the skew pantograph, a specific mechanism that can be used to scale triangles.

“The first step in our method is to triangulate the surface of the object”, explains Broeren. “Next, a tiling algorithm replaces each of the triangular faces by pantograph mechanisms in such a way that collisions are avoided when scaling. This makes it possible to scale any surface with one degree of freedom, meaning the movement takes place in the same plane as the object surface. Theoretically, we can scale structures from their fully expanded configuration down to a single point.”