Applications Of Interval Arithmetic In Geometric Modeling

With the population and application of computer, CAD and CAM is being enhanced quickly. Geometric modeling has a long history in the CAD/CAM. No doubt, the geometric modeling technical development and the application level of CAD has becomes one of the standard to measure the modernization of the scientific and the technique and its middle question is the representation of geometric model in the computer.. This representation is not only convenient for dealing with model in the computer but also satisfies requests of the representation and geometric design. Furthermore, it also is convenient for changing between information of shapes with figure of products. According to numerical value of designed model and some methods of building the foundations of curves and surfaces, it builds the mathematic models. And we can have much information about points on the curves and surfaces. We can find out the local and full characters of models that we defined.Be limited to some requirements, the product'shape which will be designed couldn't be made up with simple surface or curve. And the freeform surface or curve are combined by combinatorial surface or curve to describe. Most time, researcher put curve segment or surface patch together to express complicated object.This paper discusses some applications of interval arithmetic in geometry modeling. The representation for the boundary of curve is the commonest in geometry modeling. But this method is short of robustness Now point,line and surface are perfect and this representation is used singularly in the computer. Only a little part of subjects is represented accurately in computer because figures are float and this representation is inaccurate. For stability, interval arithmetic androunded interval arithmetic are introduced. Now, those applications are discussed in these four chapters seriously.In the first chapter, after a brief view of the role of solid modelers in CAD and CAM, we explain the importance pf robustness for curved or surfaced solid models. In a floating computer Environment, there are always some defects in the systems when computing. In order to get rid of this kind of problem, interval Bézier curves and interval B-splines curves and their properties are also discussed. We generalize interval arithmetic in the space and based on two representation of tolerance zone of basic geometric element-point (represented by a rectangle and a circle respectively), algorithms are presented for computing the tolerance zones of lines, circles and geometric transformation such as symmetry and rotation. Then we introduce the definition of ball Bézier curves and its properties: end interpolation,affine invariant,convex hull,de Casteljau algorithm,subdivision and degree elevation and reduction.Implicitness and parameterization curves are important curves in the geometric modeling. Now we use parameterization curves of degree 3 and double 3. Free curves and surfaces are defined by using parameterization foundation and these methods have advantages as follows: first, it can control shapes of curves and surfaces more freely. Second, when computing each control points, it can have least workload. Third, it do not control size of variables so is convenient for users to deal with curves in higher dimensional and we do not worry about properties and shapes in lower dimensional. Fourth, size of variables is limited so we do not define boundary of parameters. Fifth, it can simplify workload of computation. At the same time, implicitness curves have more degree of freeness, and at the same condition implicitness curves have more methods to control shapes of geometry modeling and can approximates all kinds of curves and surfaces almost.So in chapter two, we mainly introduce some applications of interval arithmetic in implicitness and parameterization of curves and surfaces that are important problems in the geometry modeling. Curve approximation is to use lower and simpler curve to approximate the requested curve. These problems have theoretical and practical importance in the computer graphics and geometric modeling. This chapter is composed of two parts. According to the concepts of implicitness and parameterization, the first part introduces a new concept called interval implicitness of rational curves that is, we find an interval algebraic curve with lower degree to bound a given parametric rational curve with higher degree. Furthermore, we subdivide rational curves and then to make interval implicitness of these curves respectively. This is introduced detailed as follows: Firstly we give the definitions of interval algebraic curve and its width: Interval algebraic curve is defined as follows: And [f₍ij0] is interval and [ 0 ] denotes zero interval. interval algebraic curve's width is: Now having this base, interval implicitness of rational curves is detailed as follows: To look for an interval algebraic curve of degree m ( m < n) that includes P ( t )( P ( t ) in the interval algebraic curve) and the interval algebraic curve's width reach the least.We attain the theorem via analyzing the problem:Theorem 1 including theoremThe full condition that an interval algebraic curve of degree m (m < n) includesP ( t ) is : So the problem is changed to an optimization problem: To solve the optimization problem, we can have an interval algebraic curve that includes rational curve. In the second part .we extend interval implicitness of curves to surfaces case. We find an interval algebraic surface with lower degree to bound a given parametric rational surface with higher degree. We mainly focus on rectangle Bézier surface patch. In the same time, we introduce the definition of interval parametric. This problem is to find a interval Bézier curve that approximates an algebraic curve and the interval Bézier curve holds G at the two end points. Similar to the process of interval implicitness, we have the process of interval parametric:Firstly: we give the definition of interval Bézier curve of degree m :Theorem 2 including theoremThe full condition of An interval Bézier curve of degree three includes an algebraic curve and holding G at the two end points is :So the problem is changed to an optimization problem: Thus we can have an interval Bézier curve of degree three.In chapter three, the important question is the application of interval B-splines in the curve reconstruction. Reverse engineering is an important problem of geometry modeling. Reverse engineering is an important technology in realizing the duplication and modifying of freeform surface parts, which includes measuring, modeling, manufacturing and inspecting of freeform surface. It's help to change the design mode from draft to models in conventional CAD systems. It's a new approach for rapid prototype manufacture. The tusk of reverse engineering is to get continuous surface. Curve reconstruction from data points is an important problem in reverse engineering. In the third chapter , we review on curve reconstruction's methods and that several typical method are introduced and its'apply scope, advantage, and weakness are analysis at the same time. One part of curve reconstruction is to introduce a new algorithm for reconstructing a piece of curve from a strip-shaped point cloud and to construct a piece of interval B-spline curve whose two boundaries fits the two boundary point series. The centric curve of the interval B-spline curve can be taken as the reconstruction curve from the strip-shaped point cloud.