The Implicit Algebraic Curves In Cagd Nature And Applied Research

In Computer-Aided Geometric Design field, two forms defined curves/surfaces are well known: parameter form and implicit one. Parameter form becomes mainstream of geometric design because of characteristic such as: simple construction, easy computation, etc.. While near 20 years research illustrates implicit form has advantage that parameter one hasn't. In this paper only the application and properties of implicit curves in CAGD have been researched. The main results are as follows:Firstly the researching status quo, mainstream and tend of implicit curves are summarized in this paper. Based on these near 20 years the applicant of implicit curves in CAGD and the characteristics of all sorts of implicit one are summarized and evaluated.In this paper the G2 -continuous blending conditions of two algebraic curves implicitly defined based on the construction of algebraic curves that interpolate the geometric constraints are presented. The theoretical analysis and the experimental results demonstrate that cubic algebraic curves having G2-continuous blending still have a freedom to fit the shape of the curve.G2-continuous closed curves are constructed through interpolating the vertices of a square using cubic implicit algebraic curves by Gn-1-functional spline idea in this paper. In virtue of this ways, the cubic curves implicitly defined are constructed and a freedom is still had to adjust the curve shapes without introducing any reference points. The experimental results also illustrate the good design effect of this approach.Based on parameter-parameter surfaces intersection curve, the differential geometric properties of implicit-implicit ones are in detail in the paper. These properties include tangents, curvature vector and higher-order derivative vector in cases of transversal intersection and tangent intersection. These results will be useful to the next research of space implicit curve in geometric modeling.