Geographic Coordinate System And Ruled Surface Based Quadric Surface Parameterization

In computer graphics and computer aided design, we need to observe the design works from different angles. Therefore, we must put the object in 3D space, where we can observe from all directions and show our design work. In this way, the design method of 3D graphics has been widely used in scientific research and engineering technology.This paper introduces a science topic which occurs in a 3D surface construction in the process of extending parameterization problem, and this paper puts forward a method based on the geographical coordinates for the ellipsoid parameterization method and ruled surface for hyperboloid surface parameterization method.We have known from very early that the motion of points can be into a line, the lines can move into a surface. In a three dimensional space, we can use a set of three dimensional coordinates to represent. A line is a set of ordered points, which are some order relations between the point and the point. A similar principle can be extended to the three-dimensional space, we can use equations to express the surface, and the equations represent the relationships between the curve and the other curve. The above methods can be easily implemented in mathematics, then, how to represent the line and surface of three-dimensional space in the computer system is the focus of the study of computer graphics.The curve or surface has many forms of expression in the computer. They can be divided into three categories roughly, explicit, implicit and parametric representation. The advantages and disadvantages of these three representations will be described in detail in this paper. Here we give a brief description of the implicit representation of curves or surfaces in computer geometry modeling system as a very simple equation form. By using this equation, we can easily determine whether the point is on the 3D curve or surface, or in the outer surface of the 3D curve or surface. If the intersection of the two geometric objects is required, the implicit representation of the curve or surface has a very fast speed and accuracy. The parametric representation of a curve or surface is a set of equations relating to the parameters. Given a set of parameters, we can quickly calculate the three-dimensional coordinates of the points corresponding to this group of parameters. So in the practical application, the implicit form and the parametric form of the 3D curve or surface are very important, so the research field of this research field is produced. The process of transforming a surface from an implicit form to a parametric form is called parameterization. At present there have already exists many kinds of methods about the curve parameterization, such as uniform parameterization method, the accumulated chord length parameterization method and quadratic precision parameterized method. The curve parameterization method has been developed. Because of its complexity, surface parameterization has not yet appeared a perfect method, but because of its important application significance, many researchers have been working and studying in this area.At present there are many methods to compute the surface parameterization. For example, finite element method, convex combination mapping method, mean coordinate method, boundary mapping method, discrete conformal mapping, discrete integral mapping method, etc.. Most of these methods are based on the transformation of the implicit equation of the curved surface. However, these traditional methods have not been able to solve the problem of the ambiguity of the pros and cons of the arc on the ellipsoid, and the crack problem of the non-single-connected surface. After the article, we will detail the two issues. This paper presents a new method which can solve these two problems, the method is based on standard quadratic implicit surface equation, under different conditions of ellipsoid and hyperboloid of one sheet, we constructed respectively effective parametric curves, selected to meet the requirements of the interval parameters. Finally, we get the parametric equation and the parameter range. The ellipsoid surface parameterization is based on geographic coordinates; parameterization of hyperboloid of one sheet is based on the decomposition of a ruled surface. Experiments show that the new method effectively avoids the ambiguity of the pros and cons of the arc, and ensure the continuity of parametric surface patches. Consequently, it has a better application value.