Optimal Rational Parameterization Of Quadratic Algebraic Surfaces

Surface is the key point in Computer graphics and Computer aided geometric design.There are two methods to represent surface: implicit representation and parameterrepresentation. Parameter representation has been widely used in computer graphics, becauseit has the characteristics of simple structure and easy to display.The effect of surface rendering quality directly affected by parameterized algorithm. Oneof the most commonly used parametric method is rational parametric. The study to parametricsurface turn to the study of rational parametric equation.In the area of today’s geometric modelling. The research of algebraic curve parametrichas made significant results. But for algebraic surface, the parameterized computation is morecomplex and more heavy, so its rational parametric study has not progressed. The traditionalrational parametric study is based on angle，but with the improvement of material level andthe high-speed development of science, people demand for appearance modelling is becomingmore and more intense. Engineering manufacturing, medical research, costume design in theareas of rapid development, designers hope to exploit the optimal parametric equation of themanufacturing or design the products more vividly, beautifully. Although traditionalmathematical methods, such as plane, sphere, cylinder, cone surface shape of the rules of thesurface, but it is difficult to display in real life, such as cars, ships, aircrafts, all kinds ofshapes, irregular surface. Parametric method in free curved surface is more convenient.For these reasons above, this article will focus on the research of quadratic algebraicsurface of rational parametric problem. Based on the optimal rational parametric standard ofalgebra curve, put forward the optimal rational judgment to algebraic surface parameterizedstandard. And according to the standard, obtains the optimal parametric method, to determinethe optimal rational parametric equation, to solve the geometric parameters of optimalparametric problems in modelling. The method can greatly improve the working efficiency ofthe computer numerical control, also is a great improvement on aesthetics, the method to drawthe surface more smooth and rounded. And put forward also the parameter of the method oftheory research and application in the future development to lay the foundation.Based on the optimal rational parametric standard of algebra curve, by the2d extended to3d. Uniform plane curve segment is the arc length of parameterized extensions for the space surface area of uniform parameterization. Through the mapping to the triangle in theplane of the surface of the optimal parameter, making the space curved surface piece also getthe optimal parametric. Analogy parameter optimal parametric curve method, the use ofsurface triangle, the ratio of maximum and minimum area and1to determine the relationshipbetween each small triangle surface area is uniform, whether it is the optimal method ofparameterized. A lot of experimental data show that the method is high accuracy, adaptive,and is a kind of rational parametric method with more efficient.