Research On Some Problems Of Self-intersection In Offset Curves And Surfaces

The 21st century is the era of rapid development of information science and technology, businesses have shown a strong dependence on modern technology, but it also reflects the high demand, computer-aided geometric design (CAGD), as an information integration, data integration, and an important tool for product development, plays an important role in this era of big data, for example, in the study of NC machining tool, path linear design, luggage and other shell design, automotive, aerospace and other forefront areas are very important practical application. Offset phenomenon is the core of the theory of curves and surfaces of these widely supported, especially in the special treatment and removal of technical control point image, Rhino software modeling process, sometimes the complexity of the phenomenon self-intersection situation occurs.The offset curves and surfaces theory, equivalent radius offset and variable radius offset are two basic types. The first category is equivalent radius offset curves and surfaces. Equivalent radius offset is relatively unique, because it is a fixed constant d as the offset distance. In the theoretical deduction or derivation, the impact of the fixed constant is generally not large on function expressions. In the simulation, the influence of the fixed constant to work is sometimes just a question of image proportion. Therefore, the researcher attached great importance to the reasonable equivalent radius offset, but also because of its particularity has promoted the depth of research to some extent The second category is variable radius offset curves and surfaces. Because the offset distance becomes a very flexible real function d(t), this flexibility does not seem to the diversity, which is not expected by people, because both theory deduction and the experimental simulation really bring difficulties and complexities, even when partial distance comes to unreasonable situation, CAD software can hardly run operation, therefore, unreasonable situation must have the reasonable processing.This article studies offset curve and surface from 2D curves self-intersection removal and three-dimensional surface removal, to achieve self-intersection removal of curves and surfaces by choosing basis functions, creating self-intersection equation, solving self-intersection equation and determining location of the intersection etc. The main contents and results are as follows:Firstly, the thesis analyzes whether the two intersect curves regular, in order to determine the non-regularity condition of intersection curve. For the study of offset curve intersection problem in the two-dimensional space, when two free curves C(u) and C(v) occur to intersect, there will be the equation C(u)-C(v)=0 at the intersection point. The core idea of elimination algorithm is to eliminate parameter items, if I(u,v)=C(u)- C(v), ultimately, transferring goal equations into the reduced-order equations and using algebraic decomposition method (u -v)(?)(u, v)=C(u)-C(v), by conversion, so that goal equations and lower-order equations finally establish an equivalent relation, just by solving the equation (?)(u, v)= 0, the result of the self-intersection equation would be obtained, and also reach the purpose of intersection removal.Secondly, based on the study of offset curves self-intersection, exploring the offset surfaces self-intersection phenomenon in the three-dimensional space. From the front experience of offset curve self-intersection issues, it can be inferred that the intersection of offset rational surfaces S(u,v) and S(r,t) also belongs to popularization of curves intersection, because the differences of self-intersection between offset surfaces and offset curves. On the basis of self-intersection removal of the offset curves, there is an additional parameter constraint [0,1], and then will have an effective promotion under the Bezier primary function.Finally, popularizing the two-dimensional, three-dimensional curves or surfaces deeply, obtaining removal self-intersection equations of high-dimensional space of intersection surfaces, and giving a theoretical deduction and certification process of multi-dimensional equations.The presentation of the elimination algorithm solves effectively the self-intersection problems of offset curves and offset surfaces, makes up the defects of removal intersection with a disk scanning algorithm, improves the theoretical system of removal intersection in the research of offset curves and offset surfaces, popularizes the practical ranges of removal intersection by the elimination algorithm, and broadens the applied scopes of removal of curves self-intersection and surfaces self-intersection.