Curvature Sensitive Mesh Generation
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.
