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M. Hensel, G. Nawratil, F. Rist, Z. Li:
"Invertible Paradoxically Loop Structures for Transformable Design";
Computer Graphics Forum, 39 (2020), 2; 261 - 275.

We present an interactive tool compatible with existing software (Rhino/Grasshopper) to design ring structures with a paradoxic
mobility, which are self-collision-free over the complete motion cycle. Our computational approach allows non-expert users
to create these invertible paradoxic loops with six rotational joints by providing several interactions that facilitate design
exploration. In a first step, a rational cubic motion is shaped either by means of a four pose interpolation procedure or a motion
evolution algorithm. By using the representation of spatial displacements in terms of dual-quaternions, the associated motion
polynomial of the resulting motion can be factored in several ways, each corresponding to a composition of three rotations. By
combining two suitable factorizations, an arrangement of six rotary axes is achieved, which possesses a 1-parametric mobility.
In the next step, these axes are connected by links in a way that the resulting linkage is collision-free over the complete motion
cycle. Based on an algorithmic solution for this problem, collision-free design spaces of the individual links are generated in a
post-processing step. The functionality of the developed design tool is demonstrated in the context of an architectural and artistic
application studied in a master-level studio course. Two results of the performed design experiments were fabricated by the use
of computer-controlled machines to achieve the necessary accuracy ensuring the mobility of the models.

http://dx.doi.org/10.1111/cgf.13928