### Curvature Sensitive Meshing

In 2-D the medial axis is the locus of the centre of an inscribed disc of maximal diameter as it rolls around the domain interior expanding and contracting to maintain contact with the domain boundary. The combination of the medial axis and the radius function, which describes the radius of the inscribed disc at any point on the medial axis, is known as the Medial Axis Transform (MAT). In 3-D the equivalent construction is the locus of the centres of all inscribed spheres of maximal diameter. This is also known as the medial axis, though perhaps the medial surface would be a more appropriate description.

The medial axis captures the geometric proximity of the boundary elements in a simple form and therefore provides a complimentary representation of physical objects in computer aided design systems. It is obvious that the effectiveness of the medial axis to capture an object's geometric characteristics influences its ability to serve these purposes, e.g. meshing, features recognition, object decomposition, path planning etc.

Currently there are various ways to construct the medial axis topology. Typically these algorithms generate excessive points such that the medial axis patch is geometrically over approximated, i.e. the number of generated points, which define a particular medial axis patch, is more than what is practically required. Hence it appears that an algorithm for the construction of medial axis geometry, which will be carried out after the topological structure has been constructed, is necessary for medial axis patch approximation. With an efficient algorithm to successfully approximate the medial axis, the subsequent processes will run more effectively.

## Strategy:

## Stage 1: Completed

Develop an adaptive curvature-sensitive mesh generator for ordinary surfaces. For an ordinary surface in 3D space, it has one, or any combination, of the following natures: elliptic, parabolic, hyperbolic and planar. The differences of these surface natures are based on the magnitude and directions of the curvature vectors. The working principle of this mesh generator lies on the first and second fundamental forms of the surface.

##

Stage 2: Completed

Develop a set of theories and formulae so that the geometric properties of the medial axis of 2D planar object can be obtained. Derived from the equal distance criterion of medial axis, this set of theories and formulae is able to generate all the necessary geometric information, such as the tangent and curvature vectors, of a particular point at the medial axis.

## Stage 3: Completed

Extent the theories and formulae found in stage 2 to the mid surface of 3D solid object. All the geometric information required to do curvature-sensitive meshing on ordinary surfaces is equally essential for mesh generation on the mid surfaces. Some principles found in stage 2 can be extended to the mid surface of 3D solid object, and the coefficients of the first and second fundamental forms of the mid surface have been derived.

## Stage 4: Completed

Combine all the results to produce an adaptive curvature-sensitive mesh generator for mid surface. By dropping all the equations and functions of stage 3 into the mesh generator developed at stage 1, the mesh generator is able to do adaptive curvature-sensitive meshing on mid surface.