#### Date of Award

1998

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Computer Science

#### First Advisor

Sukhamay Kundu

#### Abstract

The choice of a proper parametrization method is critical in curve and surface fitting using parametric B-splines. Conventional parametrization methods do not work well partly because they are based only on the geometric properties of given data points such as the distances between consecutive data points and the angles between consecutive line segments. The resulting interpolation curves don't look natural and they are often not affine invariant. The conventional parametrization methods don't work well for odd orders k. If a data point is altered, the effect is not limited locally at all with these methods. The localness property with respect to data points is critical in interactive modeling. We present a new parametrization based on the nature of the basis functions called B-splines. It assigns to each data point the parameter value at which the corresponding B-spline $N\sb{ik}(t)$ is maximum. The new method overcomes all four problems mentioned above; (1) It works well for all orders k, (2) it generates affine invariant curves, (3) the resulting curves look more natural, in general, and (4) it has the semi-localness property with respect to data points. The new method is also computationally more efficient and the resulting curve has more regular behavior of the curvature. Fairness evaluation and knot removal are performed on curves obtained from various parametrizations. The results also show that the new parametrization is superior. Fairness is evaluated in terms of total curvature, total length, and curvature plot. The curvature plots are looking natural for the curves obtained from the new parametrization. For the curves obtained from the new method, knot removal is able to provide with the curves which are very close to the original curves. A more efficient and effective method is also presented for knot removal in B-spline curve. A global norm is utilized for approximation unlike other methods which are using some local norms. A geometrical view makes the computation more efficient.

#### Recommended Citation

Lim, Choong-gyoo, "A Universal Parametrization in B-Spline Curve and Surface Interpolation and Its Performance Evaluation." (1998). *LSU Historical Dissertations and Theses*. 6686.

http://digitalcommons.lsu.edu/gradschool_disstheses/6686

#### ISBN

9780591904734

#### Pages

76