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Closed-form expressions for the approximation of arclength parameterization for Bezier curves

Madi M.

International Journal of Applied Mathematics and Computer Science

2004

Vol. 14, no 1

33-41

In applications such as CNC machining, highway and railway design, manufacturing industry and animation, there is a need to systematically generate sets of reference points with prescribed arclengths along parametric curves, with sufficient accuracy and real-time performance. Thus, mechanisms to produce a parameter set that yields the coordinates of the reference points along the curve Q(t) = {x(t), y(t)} are sought. Arclength parameterizable expressions usually yield a parameter set that is necessary to generate reference points. However, for typical design curves, such expressions are not often available in closed form. It is thus desirable to find efficient ways to compensate for this lack of arclength parameterization. In this paper, several methods for approximating arclength parameterizations are studied. These methods are examined for both accuracy and real-time processing requirements. The application of generating reference points uniformly spaced along the paths of several curves is chosen for the illustration and comparison between the presented methods.

An interactive modification data structure for 3D surfaces

Madi M., Walton D.

Machine Graphics and Vision

2003

Vol. 12, No. 3

293-310

The design and display of 3D models on a computer screen is usually interleaved by a series of data manipulations. Data are usually sets of 3D points that make up polygonal patches in the composition of polyhedral models. Manipulations are the transformation operations that are applied to points to facilitate design and visual understanding of the graphical models. To further facilitate and speed up the design phase, methods are proposed to interactively segment and modify selected sub-surfaces, thereby limiting the number of patches that need to be manipulated for computationally cheaper and faster results. This paper illustrates how polyhedral surfaces can be organized into special data structures to facilitate rapid selection of vertices, and how those same surfaces can be segmented into sub-polyhedra for zooming and vertex manipulation during design, and can then be re-introduced as modified segments into the original structure. Algorithms and visual examples are also provided to support the work.