| In many applications of computer aided geometric design, the curves with satisfyingshape should not constitute any singularities and undesirable inflection points. In addition,convexity property plays indispensable role in geometric modeling. These geometricproperties have direct impacts on various aspects of shape design such as the dynamicsperformance, algorithm complexity and feasibility in manufacture. Therefore, to avoidpotential risk in shape design, it is of vital importance to analyze and predict the shape ofparametric curves, which can be seen as one of most fundamental tools for geometricmodeling. In the literature, there are three kinds of classical methods for shape analysis ofparametric curves, including the Affine Invariant Method (abbreviated, AIM) proposed by Suand Liu, Stone and DeRose’s Geometric Characterization Method (abbreviated, GCM) basedon the position of the control points, and Ye’s method based on the theory of envelopes andtopological mappings, which will be referred to as Ye Method in what follows. Althoughmany classic results have been obtained in the study of shape features concerning planarparametric cubic curves and their rational forms, very few practical results have beenpresented with regard to algebraic polynomial parametric curves with higher degree andnonalgebraic parametric curves.The purpose of this dissertation is to study the geometric characteristics of parametriccurves with differential shapes, criteria for distinguishing local convexity from globalconvexity, and shape analysis of some planar quartic polynomial curves as well astrigonometric parameter curves. The main contribution lies in the following aspects:1. The shape diagrams of Ye Method are classified into three kinds of Bézier types andthree kinds of B-spline types, and the relationship of shape diagrams of AIM, GCM and YeMethod is ascertained. Moreover, the distribution regions corresponding to local convexityand global convexity of all these shape diagrams are partitioned.2. To overcome the shortcomings of the aforementioned methods, we propose a novelmethod, called Characteristic Space Method (abbreviated, CSM), to determine the shapefeatures of parametric curves. The main idea of our method is to construct characteristic cones(including cusp cones and two end loop cones) in terms of three side vectors in the geometriccharacteristic equations of parametric curves. The characteristic cones and their tangentplanes partition the3-dimension space into different characteristic regions according to shapefeatures, which is called the characteristic space of parametric curves. By CSM, we obtainsome useful theorems on characteristic spaces of cubic Bézier curves and cubic B-spline curves. It is shown that the shape diagrams of GCM and Ye Method can be derived fromcharacteristic spaces by virtue of planar slices, which are vertical to one of the axes; while theshape diagrams of AIM can be obtained by nonplanar section through the characteristic space.The advantages of our method are three-fold: Firstly, CSM does not rely on the parallelassumption of control side vectors. Secondly, the shape features of parametric curves can becompletely determined by the position of the characteristic point in a characteristic space,except for the trivial case in which the four control points are collinear. Last but not least, thecomputation of the characteristic point in CSM needs only three determinants about thecontrol side vectors, and the region determination of the characteristic point involves onlyplanes and three cones. Therefore, this method is more suitable to the automatic determinationof shape features by computer programs.3. Using Ye Method, we obtain the shape conditions of planar rational cubic Béziercurves and four classes of parametric curves with shape parameters, including planar quarticpolynomial curves, spline curves, planar trigonometric Bézier curves and C-B-spline curves.We find that the structures of these shape diagrams are very similar to those of cubicparametric curves.Furthermore, we also discuss the influences of shape parameters on the associated shapediagrams. The obtained results can enable the user to place control points or choose shapeparameters such that the resulting curve is globally or locally convex, possessing wantedsingularities or inflection points, or with the desired shape. In addition, some special shapediagrams are also discussed when the shape parameters beyond the restricted interval.4. For the above mentioned parametric curves, we construct their cusp cones and endloop cones. Finally, we investigate the impact of shape parameters on characteristic space. |
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