Beta-spline curves : properties and computations
- Bangkok : Asian Institute of Technology, 1989
- 77, [56] p. + 1 online resource
- Asian Institute of Technology. Thesis ; no. CS-89-11 .
A thesis submitted in partial fulfillment of the requirements for the degree of Master of Engineering
Thesis (M.Eng.) - Asian Institute of Technology, 1989
The Beta-spline developed by Barsky is to combine uniform cubic B-splines with geometric continuity using the unit tangent vector and curvature vector. Two shape parameters called bias and tension respectively were introduced to take care of the geometric continuity constraints. The effects of these two parameters on the shape of the curve have been studied. Uniformly-shaped Ý-spline has fixed shape parameters which affect the curve over the whole length. For a more 2local3 sense, continuous-shaped Ý-spline is formed by introducing quintic interpolating polynomial between the distinct Ý values at each knot. Some 2kinks3 is created in case of wildly varying adjacent Ý values. This is not a desired situation in Computer Aided Geometric Modeling. Later, Goodman gave a general definition of Ý -splines. The basic properties of convex hull and variation diminishing have been shown. An explicit formula for cubic Ý-splines has been given. Starting from this definition and the explicit formula, a B- Ý spline concept was suggested for computational efficiency thanks to its recurrence relation. The computation scheme and implementation were also given. Analogous to Ý-splines, discrete Ý-splines with the technique of subdivision and the knots insertion on the Ý-spline curves have been studied, based on the above general definition of B-spline. A computation scheme of the above two methods was discussed.