Planar to spatial geodesic grids

At the Center for Geometry and Computational Design (GCD) (Institute for Discrete Mathematics and Geometry) at TU Wien, Musialski and his team developed a method that can be used to calculate what the flat, two-dimensional grid must look like in order to produce exactly the desired three-dimensional shape when it is unfolded. “Our method is based on findings in differential geometry, it is relatively simple and does not require computationally intensive simulations,” says Stefan Pillwein, first author of the current publication.

Suppose you screw ordinary straight bars together at right angles to form a grid, so that a completely regular pattern of small squares is created. Such a grid can be distorted: all angles of the grid change simultaneously, parallel bars remain parallel, and the squares become parallelograms. But this does not change the fact that all bars are in the same plane. The structure is still flat.

The crucial question now is: What happens if the bars are not parallel at the beginning, but are joined together at different angles? “Such a grid can no longer be distorted within the plane,” explains Przemyslaw Musialski. “When you open it up, the bars have to bend. They move out of the plane into the third dimension and form a curved shape.”

At the Center for Geometry and Computational Design (GCD) (Institute for Discrete Mathematics and Geometry) at TU Wien, Musialski and his team developed a method that can be used to calculate what the flat, two-dimensional grid must look like in order to produce exactly the desired three-dimensional shape when it is unfolded. “Our method is based on findings in differential geometry, it is relatively simple and does not require computationally intensive simulations,” says Stefan Pillwein, first author of the current publication.

You can read more in the original¬†paper¬†(this version is adapted and abridged from Source Vienna University of Technology. “The mathematical magic of bending grids.” ScienceDaily. ScienceDaily, 24 August 2020.).

Pillwein, S., Leimer, K., Birsak, M. and Musialski, P., 2020. On elastic geodesic grids and their planar to spatial deployment. arXiv preprint arXiv:2007.00201.