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In this paper, the method of finding the line constrained between two quadratic Bezier curves and also finding the line constrained between a quadratic Bezier curve and a circle is presented. The application of the line constrained can be used in the construction of railway tracks between any obstacle or rolling a ball to the other side of a wall in a way that it just touches the wall. The method used is by using equal root properties of a quadratic equation in order to find one point where the line touches the curve. The work examples involved different curve orientations and different circle positions. Mathematica software is used to compute the solutions for the line constrained and present the solutions graphically. By using the method proposed, the number of intersection points obtained is used to determine the number of lines constrained between two curves and between a curve and a circle. The conclusion on whether all lines are acceptable to be considered as the line constrained are depending on the application of the line. [ABSTRACT FROM AUTHOR]

Branching is a process of connecting two adjacent contours called base and branch contours where the numbers of sub contours for each of both contours are different. All points between the contours have to be correctly paired to avoid a twisting surface. In contours connection, continuity is an issue to ensure the surface fit to the required smoothness. Therefore, two main steps to be focused in surface reconstructions are branching technique and surface continuity. This study used composite contour generation as the branching technique which generates an intermediate contour as combination of modified base and branch contours. Although the inserted new contour will disturb the position of original contours, the continuity can be preserved if the fitted surface is Beta-spline. Beta-spline is built with G2 continuity condition, and the changes of the data points position will not affect the continuity. Hence, surface fitting using Beta-spline will simplify the reconstruction process. The result shows several images can be generated using Beta-spline where composite contour is used for branching problem. The result then compared with the original image and it is shown that surface reconstruction using Beta-spline can give a good result especially for the branching case. [ABSTRACT FROM AUTHOR]