The Sobolev gradient method is a powerful tool for geometric modeling. We treat the problem of constructing fair curves by minimizing a fairness measure subject to geometric constraints. The measure might include curve length, curvature, torsion, and/or variation of curvature. The constraints may include specified values, tangent vectors, and/or curvature vectors. We may also require periodicit...

Plasma enhanced techniques are widely used for synthesis of carbon nanostructures. The primary focus of this paper is to summarize recent experimental and theoretical advances in understanding of single-wall carbon nanotube SWNT synthesis mechanism in arcs, and to describe methods of controlling arc plasma parameters. Fundamental issues related to synthesis of SWNTs, which is a relationship bet...

The registration of 3D form features is essential to the supporting of reverse shape design processes. Extracting an editable shape feature from unordered data points is notoriously hard. We present a new method to extract a shape instance plus design parameters from a ridge structure contained in a freeform surface, where the ridge has no predefined path, and can have varying height and width ...

The problem considered here is to determine the shape of a symmetric twodimensional plate so that the drag of this plate in infinite cavity flow is a minimum. With the flow assumed steady and irrotational, and the effects due to gravity ignored, the drag of the plate is minimized under the constraints that the frontal width and wetted arc-length of the plate are fixed. The extremization process...

We develop a numerical method based on Sine functions to obtain an approximate solution of a one-dimensional Cauchy singular integral equation (CSIE) over an arbitrary, smooth, open arc L of finite length in the complex plane. At the outset, we reduce the CSIE to a Fredholm integral equation of the second kind via a regularization procedure. We then obtain an approximate solution to the Fredhol...

We apply the penalty method to the curve straightening flow of inextensible planar open curves generated by the Kirchhoff bending energy. Thus we consider the curve straightening flow of extensible planar open curves generated by a combination of the Kirchhoff bending energy and a functional penalising deviations from unit arc-length. We start with the governing equations of the explicit parame...