A family of methods for the efficient update of second order approximations of a
constraint manifold is proposed in this thesis. The concept of such a constraint manifold
corresponds to an abstract space prescribed by implicit nonlinear constraints,
which can be a set of objects satisfying certain desired properties. This concept has
a variety of applications, and it has been successfully introduced to fabrication-aware
architectural design as a shape space consisting of all the implementable designs.
The local approximation of such a manifold can be first order, in the tangent space,
or second order, in the osculating surface, with higher precision. For a nonlinearly
constrained manifold with rather high dimension and codimension, the computation
of second order approximants (osculants) is time consuming. In this thesis, a type
of so-called quasi-Newton manifold exploration methods which approximate the new
osculants by updating the ones of a neighbor point by 1st-order information is introduced.
The procedures are discussed in detail and the examples implemented to
visually verify the methods are illustrated.
|Date of Award||Aug 2011|
|Original language||English (US)|
- Computer, Electrical and Mathematical Science and Engineering
|Supervisor||Helmut Pottmann (Supervisor)|