Research On The Rendering Algorithm Of Complex Implicit Curve

Because of the requirement and advantages of Implicit Curve/Surface, ImplicitCurve/Surface is receiving more and more attention in Computer Aided GeometricDesign (CAGD) and Computer Graphics. Implicit Curve/Surface has difficulties inrendering, but it still has advantages that can not be substituted: First, teoreticallyspeaking, implicit Curve/Surface is able to maintain all of the characters of ParametricCurve/Surface and has more freedom. Second, Implicit Curve/Surface provides theenough flexibility is able to fit any complex curve and surface. Third ImplicitCurve/Surface has the advantage to quickly identify whether the point is inside oroutside or on the curve/surface. Fourth Implicit Curve/Surface not only has the thecompact succinct expression form, but also is closed in intersection, summation,subtraction and offset operation., and it is very important in entity modeling system. Ifa geometry design system can not contain Implicit Curve/Surface, then the aboveoperation surpasses its scope. So the research on Implicit Curve/Surface renderingalgorithm has important practical meaning.This article analyzes the existing Interval Arithmetic and Subdivision algorithm,And makes further research on rendering of complex Implicit Curve, because of therequirement, research takes out a definition of trigonometric function, index function,logarithm function operating principle. This definition is related with Subdivisionalgorithm, applying these definition and Subdivision algorithm render the complexImplicit Curve cause by 3x+1 promotion function, and then find the fixed points on thecurve. Now briefly introduce the main content of this article according to the chapterorder: In the 1st chapter, analyzes the advantage and disadvantage of Implicit Curvecompare with Parametric Curve, and brief introduce the source and definition ofInterval Arithmetic, Subdivision algorithm, and 3x+1 promotion function, finallymakes a summary of this article.In the 2nd chapter, traverses the Interval Arithmetic, and realizes it on computer,takes out an algorithm combines with Interval Arithmetic and Subdivision, it is mainalgorithm of this article. Renders and compare ten typical implicit polynomialfunctions curve by this algorithm and direct algorithm, lists number of pixels and costof this tow algorithms, analyzes the advantage and disadvantage of this tow algorithms.In the 3rd chapter, describes the application of Interval Arithmetic in renderingtrigonometric function in detail, First carries on the analysis of the trigonometricfunction characteristic, based on this, points out that all of the trigonometric functionscan be deducted by researching on operation rules of sin(). And definite the operationrule of Interval Arithmetic by analyzing the characters of trigonometric function,operation rule of Interval Arithmetic of trigonometric function by Taylor expressionand its improvement 1 and 2. This chapter is the description of the definition process ofInterval Arithmetic of trigonometric function through effect pictures, because of thisdefinition is combined with Subdivision algorithm, so it still have to make furtherresearch to apply this definition in other field.In 4th chapter, brings out the definition of operation rules of index function andlogarithm function. Based on definition of operation rule of Interval Arithmetic rendersseveral implicit curve of 3x+1 promotion function, and finds the fix points on complexnumber area, and then lists and analyzes them. Because of the characters of IntervalArithmetic and resolving ability of computer, these points are not very precise. Articleanalyzes the characters and distributing rule represented by these fix points, but thisresult still have to make further study to prove.In 5th chapter summarizes the whole article and points out the direction of thefurther study.