Discrete geodesic parallel coordinates

Abstract

Geodesic parallel coordinates are orthogonal nets on surfaces where one of the two families of parameter lines are geodesic curves. We describe a discrete version of these special surface parameterizations and show that they are very useful for specific applications, most of which are related to the design and fabrication of surfaces in architecture. With the new discrete surface model, it is easy to control strip widths between neighboring geodesics. This facilitates tasks such as cladding a surface with strips of originally straight flat material or designing geodesic gridshells and timber rib shells. It is also possible to model nearly developable surfaces. These are characterized by geodesic strips with almost constant strip widths and are used for generating shapes that can be manufactured from materials which allow for some stretching or shrinking like felt, leather, or thin wooden boards. Most importantly, we show how to constrain the strip width parameters to model a class of intrinsically symmetric surfaces. These surfaces are isometric to surfaces of revolution and can be covered with doubly-curved panels that are produced with only a few molds when working with flexible materials like metal sheets.

On elastic geodesic grids and their planar to spatial deployment

“…On the theoretical side, a novel representation of developable surfaces using quadrilateral meshes with appropriate angle constraints [Rabinovich et al 2018] or a definition of developability for triangle meshes [Stein et al 2018] have been proposed recently. Also discrete geodesic parallel coordinates for modeling of developable surfaces were proposed [Wang et al 2019]. All these works aim at the design of developable surfaces, which, due to their isometric properties, can be fabricated from planar sheets.…”

Shape approximation by developable wrapping

“…The representations are based, e.g., on vanishing Gaussian curvature [Wang and Tang 2004], isometry to the plane [Grinspun et al 2003], or the special parameterizations admitted solely by developable surfaces. The latter parameterizations include conjugate ruled nets [Liu et al 2006;Bo and Wang 2007;Solomon et al 2012;Tang et al 2016;Stein et al 2018], orthogonal or parallel geodesic nets [Rabinovich et al 2018a;Wang et al 2019], or isometric quadrangulation of a planar domain to achieve discrete-isometric mappings [Jiang et al 2020]. Sellán et al [2020] focus on modeling developability of height fields and cast it as a convex rank minimization problem.…”

Developable Approximation via Gauss Image Thinning

“…Attention has been devoted to smooth representations of developables, e.g., via B‐spline surfaces [TBWP16, GSP19] or via parametric curves with associated envelopes of rectifying planes [BW07] or tangent planes [BR93,PW99] – as well as discrete representations, which can afford greater flexibility. Prominent discretization examples include strips of planar quadrilaterals (PQ strips) [LPW∗06, VVHSH21], ruled meshes [RSW∗07,SVWG12], discrete orthogonal or parallel geodesic nets [RHSH18,WPR∗19], as well as discrete isometries of planar checkerboard patterns [JWR∗20].…”