Research items

1. Creation of novel theory of discrete surfaces
2. Construction of discrete variational principle of discrete surfaces
3. Theory of aesthetic shapes by Klein geometry

Outline

1. (a)Extending the discrete differential geometry which usually adopts planar quadrilaterals as the shape elements, we define a class of piecewise smooth surfaces with the properties of both discrete and smooth surfaces. We introduce basic notions such as the curvature. See 1, 2 in the references.

   (b)Curvature at the singularities (folds): In the architecture, in joining several pieces (plates), it is stronger to join them on a curve rather than a line. So we define a functional which indicates the strength of the joint and define the curvature at the folds by using its critical condition. This provides a new idea for the curvature at the singularities.

   (c)We construct the theory of isometric deformation of the piecewise smooth surfaces.

   (d) We introduce a distance in the class of piecewise smooth surfaces. It may be useful, for example, to add certain quantities reflecting the difference of curvature.



2. We construct the methods of analysis such as the discrete variational principle in the class of piecewise smooth surfaces.

3. (a) We introduce the notion of “aesthetic surface” to enable the efficient design of artistic shapes from the viewpoint that it is natural to consider it in the framework of the similarity geometry, which is one of the Klein geometry. The key idea is to adopt the framework that the log-aesthetic curve, which is the basis of the aesthetic surface, is the similarity geometric analogue of the Euler’s elastic curves in the Euclidean geometry (See 3. In the references).

   (b)We construct the theory of aesthetic surfaces to characterize them as the solutions of the optimization problem of the surface shapes to enable the design of “aesthetic shapes.”

Placement in the project

The researches of this group provide the mathematical basics and the fundamental methodology for the whole project and the practical design of the structures. Namely, by defining the piecewise smooth surfaces endowed with both of the discrete and smooth surfaces, introducing the variational calculus, useful definitions of the curvatures and normal vectors at the singularities, and by constructing the theory of isometric deformations, we create and develop the novel geometry of discrete surfaces and the new variational principle, and provide them to architecture, design engineering and computational geometry. Furthermore, introducing the notion of “beauty and artisticity” to the geometry of curves and surfaces, we give the basic theory for the development of an innovative platform that enables structural design with high efficiency and low cost, endowed with beauty and artisticity, and guaranteeing security and safety. For each research problem, we collaborate with all of the research groups.

References

1. M. Koiso. Uniqueness of stable closed non-smooth hypersurfaces with constant anisotropic mean curvature, preprint. arXiv:1903.03951 [math.DG]
2. Y. Jikumaru and M. Koiso. Non-uniqueness of closed embedded non-smooth hypersurfaces with constant anisotropic mean curvature, preprint. arXiv:1903.03958 [math.DG]
3. J. Inoguchi, K. Kajiwara, K.T. Miura, M. Sato, W.K. Schief and Y. Shimizu, Log-aesthetic curves as similarity geometric analogue of Euler’s elasticae, Computer Aided Geometric Design 61(2018) 1-5,
   https://doi.org/10.1016/j.cagd.2018.02.002.