Exponential Parameterized Cubic B-Spline Curves And Surfaces - Ammad, Muhammad (2020)
The use of B-spline interpolation function for curves and surfaces has been developed for many reasons. One reason is the higher degree of continuity and smoothness. A general B-Spline is a polynomial curve and its shape is determined by the control points. To interpolate data points, various works have been done by previous researchers who studies B-Spline parameterization. In this thesis, we develop a new way for interpolating cubic B-Spline curve by taking the first and the second derivative at endpoints and only the first derivative at inner points. The proposed method is the extension in the B-spline interpolation technique of using arbitrary derivatives at end points. In developing B-spline curve interpolation method, an algorithm is presented for interpolating data points. The algorithm computes knot values for parameterization methods. These knot values are used in constructing a matrix of B-Spline basis function and derivative of the basis function. Then, we solve it for control points by using the LU decomposition method, such that the curve will pass through the given data points. Selection of proper parametrization technique is critical for curve and surface reconstruction process. The parametrization method used in this study is an exponential parameterization method with a = 0:8. The main advantage of developing B-spline curve interpolation method is that we can generate different shapes of curves by setting different direction at all data points. As an application, we applied the proposed method in curve reconstruction on a road map from given data points and driving directions, and also for path planning in autonomous vehicle with given starting and goal position.