Generalized Cubic DP Curves And Its Shape Analysis

Computer Aided Geometric Design (CAGD) developed nearly forty years. Those curvesand surfaces, e.g. Bezier, B-sp1ine and NURBS, p1ay an important role in CAGD. They aredeveloped mature in basic. But they are still insufficient, such as no encompassingtranscendent curves and surfaces，spiral curve, cycloid, arc, cone, etc. Therefore, manyscholars have studied the new type of curves and surfaces. This article introduced two shapeparameters to the cubic DP curve of in basic of Delgado and Pe a structure of DP curve,respectively, shapes, and the nature of the curve interpolation are studied, the main work is asfollows:1. We introduced two shape parameters basic on cubic DP basic function, it isGeneralized Cubic DP basic function. Then, the properties of the new basis is analyzed. Thecubic DP polynomial curves and surfaces with two shape parameters is defined using the newbasis. Not only the new curve and surfaces retain many properties of cubic DP curve andsurfaces, but also can adjust the shape by moderating shape parameters λ or μ. Based onsome given conditions can be satisfied, we can moderate the angle between the first side ofthe controlling polynomial and the tangent to the first endpoint so that the curves can beG1, G2, C1,C2continuous, when the two segments connect, which is more useful in free curveand surface design.2. The shape features of generalized cubic DP curve with shape parameters are analyzedby using the method based on the theory of envelope and topological mapping. Necessary andsufficient conditions are discussed for generalized cubic DP curves of singular points,inflection point, local and globally convexity, and obtained shape diagram of the distributionregions. The influences of shape parameters on the shape diagram are analysis.3. Because general cubic DP curve have not endpoint vector cutting qualities, so it ismore difficult if you want to use the general cubic DP curve to interpolating a lot of electrovalue point. But we found that the generalized cubic DP curve and cubic Bezier curve withtwo parameters has the close relation, and they can be mutual conversion, so, at first, we canuse the cubic Bezier curve with two parameters interpolation, and then converted to thegeneralized cubic DP curve interpolation problem.4. For the cubic Bezier curves with two parameters. A method for C2-continuity shapepreserving interpolation is given. The proposed approach of inserting two new control pointsbetween each two neighboring data points can make cubic Bezier curve with shape parameters interpolate the original control points. It is not only C2-continuity, but alsoshape preserving. Also local modification and shape of curve is adjusted by adjusting theparameters. Finally, the examples of shape preserving interpolate the Akima’s data point andthe points on the circle are given.