Smooth Surfaces via Nets of Geodesics

Fast Conformal Parameterization of Disks and Sphere Sectors

Left image source: Li Wang et al. “Structure and mechanogating of the mammalian tactile channel PIEZO2”. In: Nature 573.7773 (Aug. 2019), pp. 225–229. doi: 10.1038/s41586- 019- 1505- 8. url: https: //doi.org/10.1038/s41586-019-1505-8. 

The image shows the structure of the Piezo2 protein. Piezo1 and Piezo2 proteins are key elements in receptors for temperature and touch. David Julius and Ardem Patapoutian recieved the 2021 Nobel Prize in Physiology or Medicine for discoveries in this area. See:  https://www.nobelprize.org/uploads/2021/10/press-medicineprize2021.pdf.

Right image: to the left there is a mesh of a sphere with a source point in the bottom and three points on the top which are located symmetrically. On the right are depicted the symmetrical paths emanating from the the source and reaching the three targets.

Bottom images: Together with my brother Ben Gilat, we proved a linear algorithm for the conformal parameterization of 3-fold symmetrical meshes. The parameterization is based on a construction of a torus from 63 copies of the original mesh. The bottom images show the parameterization of the torus constructed out of 63 copies of the sphere shown above. The second image is a zoom-in of the first image. For this parameterization three symmetrical cuts of the mesh from a single point of rotational symmetry are needed. These are the lines depicted above emanating from the bottom point. This methods can be applied for example for the "interacting surface" of the Piezo2 protein where three symmetrical cuts are provided along the blades on the mesh which represents its "interacting surface".