Discrete orthogonal structures.

To represent smooth geometric shapes by coarse polygonal meshes, visible edges often follow special families of curves on a surface to achieve visually pleasing results. Important examples of such families are principal curvature lines, asymptotic lines or geodesics. In a surprisingly big amount of use-cases, these curves form an orthogonal net. While the condition of orthogonality between smooth curves on a surface is straightforward, the discrete counterpart, namely orthogonal quad meshes, is not. In this paper, we study the definition of discrete orthogonality based on equal diagonal lengths in every quadrilateral. We embed this definition in the theory of discrete differential geometry and highlight its benefits for practical applications. We demonstrate the versatility of this approach by combining discrete orthogonality with other classical constraints known from discrete differential geometry. Orthogonal multi-nets, i.e. meshes where discrete orthogonality holds on any parameter rectangle, receive an in-depth analysis. [Display omitted] • Discrete orthogonality based on rhombic mesh pairings. • Non-planar orthogonal multi-nets expanding Ivory's Theorem. • Compatibility with other classical nets leading to a wide range of applications. • Well suited for efficient optimization. [ABSTRACT FROM AUTHOR]