Smooth Connection Of Q-bézier Curver\Surface

Q-Bézier curves and surfaces are a kind of generalization of the classical Béziercurves and surfaces, which play a significant role in CAGD and computer graphics.Since they are newly generated curves and surface in recent years, there is severalresearch of the curves and surfaces. This article studies the curves and surfaces amore in-depth on the basis of the definitions and fundamental properties of the curvesand surfaces. It mainly studies the extension and the smooth connection of theq-Bézier curves. Further, in the case of curved surfaces, this paper studies the smoothconnection of q-Bézier surfaces, which contains two parameters value, in the form ofthe tensor product. Specific work is as follows:The study of extension of q-Bézier curves is on the basis of blooming andsubdivision. In terms of the1/2segmentation algorithm, a new recursive algorithm isconstructed. By extending the interval [a, b]of the q-Bézier curve to a larger interval[a, c], the control points of the q-Bézier curve after extension can be calculated by anew recursive algorithm. The new recursive algorithm and1/2segmentation algorithmis a reverse process to each other. The extension of q-Bézier curves can be divided intotwo extension styles, including the left extension and the right extension. The leftextension and the right segment as well as the right extension and the leftsegmentation are also reverse processes to each other.In the research on the q-Bézier curves’ smooth connection, the theoreticalnumerical conditions are first introduced. Then, the conditions on smooth connectionin the sense of geometric meaning are given. Consequently, we can generalize thesmooth connection of Bézier curves to the q-Bézier curves, which makes it moreintuitive. In this part, a transformation matrix operator is constructed, with which wecan change q-Bézier curves into classical Bézier curves. Further, by means of thesmooth connection conditions of Bézier curves, the smooth connection conditions ofq-Bézier curves are obtained, which gives a geometric interpretation. On this basis,another geometric conditions of smooth connection by extension algorithms andsegmentation algorithms are proposed.Finally, the problems of smooth connection on q-Bézier surfaces are studied inthis paper. A transformation matrix operator, which is an upper triangular blockmatrix，is constructed. With which the problem of q-Bézier surfaces’ smoothconnection can be transformed into the classical Bézier surfaces’ smooth connection. So the issue is resolved...