# Circular, Conical Mesh and Offset Mesh

There are two special cases in the planar quadrilateral mesh family, namely, circular mesh and conical mesh. Both meshes have some interesting geometric properties. For more information, see the paper by Liu at el.

Below are the geometric facts summarized from this paper. A circular mesh is a quad mesh that all of the vertices in a face lies on a circle; while a conical mesh is a quad mesh that all edges emanating out of a single vertex lies on a cone. These meshes both have non-trivial offset meshes that guarantees planar support structure: when the corresponding edges between two meshes constructs a face, the face is always planar. A trivial offset mesh is a scaled and translated version of the original mesh, which is the only possible offset mesh that guarantees planar support for a triangulated mesh. Offset meshes are useful in building structure from planar material. See these projects, dragonfly by Tom Wiscombe and honeycomb morphologies by Andrew Kudless for example, although some projects take advantage of the material to cope with the non-planar issue.

Offset Mesh: Dragonfly by Tom Wiscombe

Offset Mesh: Honeycomb by Andrew Kudless

In the previously mentioned paper, the author provides two different methods to construct a conical mesh, which one is more of a top-down approach and the other is more bottom-up. The top-down approach takes a mesh that is almost a conical mesh, which can be obtained by conjugate curve network, and apply an optimization process on the mesh to ensure the conical property. A conjugate curve network consists of two sets of curves that are always perpendicular, such as min/max principal curve, contour and gradient curves etc. The bottom up process applies subdivision scheme such as Catmull Clark and Doo Sabin on a rough quad mesh and optimize along the subcivision steps.

Before reading this paper, I always consider mesh and subdivision are more useful tools for animation and visualization than architectural construction. However, I am now convinced that through the research of discrete differential geometry, they can help to improve the constructability. I will test these ideas in the future.