Discrete Geodesic Nets for Modeling Developable Surfaces

Reviewer: Joseph J. O'Rourke

Developable surfaces are those that can be flattened to the plane isometrically, that is, without stretching or tearing. They play an important role in manufacturing and architecture, for example, curved glass can be constructed by rolling and bending flat glass. This paper proposes a new discrete model for developable surfaces. The new model is a quadrilateral mesh with angle conditions surrounding each vertex. When the four angles are equal, the discrete tangents are orthogonal. Thus, the model is a "discrete orthogonal geodesic net." The authors prove that "a smooth surface is developable if and only if it can be locally parameterized by orthogonal geodesics." Their net is then a discrete analog of this parametrization; "curvature line nets are a special case." The authors convincingly argue that their model is more flexible, supporting vertex-handle editing. Such editing deformations should be isometries, and toward that end they add the condition "4Q": each four-quadrilateral patch should have equal lengths of opposing sides. Then two orthogonal 4Q nets are isometric if there is a one-to-one correspondence that matches these lengths. This can then guide the deformations to be isometries, as they amply illustrate. This is computationally expensive, limiting editing to surfaces of about 1000 vertices. Much work remains.