## MEMBERS

### Program Members

With the help of support from this GCOE, I am planning to do joint research with Tim Hoffmann on the differential geometry of discrete surfaces. This work is related to applications to visualization in computer graphics and to architecture.

The discrete surfaces we consider are made by gluing planar quadrilaterals together along edges. We do not consider arbitrary surfaces. Rather, we restrict to those that satisfy an “isothermic” condition that is a natural analog of a property of smooth surfaces.

The advantage of this restriction is that it allows for the use of integrable systems theory, and gives parametrizations of the discrete surfaces that exhibit geometric properties. Because of this, the discrete surfaces are optimally elegant. This is why they are of interest in visualization and in architecture. All computer graphics are necessarily discrete at some level, because they are made with computers, and computers can only hold a finite amount of data.

Choosing isothermic parametrizations for the necessarily discrete versions of surfaces in computers is one way to find optimal choices for the data of the surfaces. Also, if an architect were to put a structure on a building or bridge, or some other structure, so that it is simulating a smooth surface, but is in fact made of a collection of small planar pieces, he would want a method for choosing those pieces that maximizes the beauty of the structure-- isothermic discrete surfaces are one possibility for this.

I have recently completed a paper with Udo Hertrich-Jeromin on discrete constant mean curvature surfaces in the space forms Euclidean 3-space, spherical 3-space and hyperbolic 3-space. In the course of doing that research, I developed an understanding of the notion of discrete isothermicity. In particular, I now understand how concircularity of vertices of the quadrilaterals in a surface, and how the cross ratio of those vertices, is used to define discrete isothermicity.

However, from recent discussions with Tim Hoffmann, who recently arrived in Kyushu University, I am beginning to see how the notion of isothermicity can be generalized in ways that I had not thought about before, and in ways that are more amenable to applications in industry. Therefore, I will continue to extend the research I did with Udo Hertrich-Jeromin by making joint research with Tim Hoffmann.

In practical terms, I will investigate discrete flat surfaces in hyperbolic 3-space with Tim Hoffmann, and also with Yoshida Masaaki and Sasaki Takeshi. Our plan is to write joint papers involving the four of us, in which we will do the following:

１）define discrete flat surfaces, and show that they have the natural property that the vertices of each quadrilateral are concircular;

２）explain how discrete linear Weingarten surfaces can be defined and use them to create a deformation that takes discrete flat surfaces to discrete constant mean curvature 1 surfaces in hyperbolic 3-space;

３）although the flat surfaces will not be isothermic themselves, we will explain how the corresponding discrete constant mean curvature 1 surfaces in the above deformation are isothermic;

４）because singularities such as cuspidal edges and swallowtails naturally appear on smooth flat surfaces in hyperbolic 3-space, we will investigate how to define discrete analogs of cuspidal edges and swallowtails and other singularities.

Perhaps the fourth objective is the most interesting, because singularities on discrete surfaces (that are any more complicated than just umbilic points) have not yet been investigated by anyone.

Furthermore, while conducting this research, we will continually make computer graphics and animations of the discrete flat surfaces, in order to explore applications to fields outside of mathematics, such as computer visualization and architecture as noted above.

A discrete helicoid

A swallowtail singularity