Research On Several Algorithms Based On Inequality Estimation In Computer Graphics

Function approximation and bounding box-based clipping are two basic problems in computer graphics,which also have wide applications in geometric modeling system and numerical simulation.This paper focuses three algorithms based on inequality estimation in computer graphics,including:(1)Improved triangular inequalities and its application in K-means clustering.A two-point Pade approximant method is presented for refining some remarkable trigonometric inequalities,including the Jordan's inequality,Kober's inequality,Becker-Stark's inequality,and Wu-Srivastava's inequality.The corresponding proofs are also provided.Numerical examples show that the new method can achieve better approximation results than those of prevailing methods.And the results can be applied for K-means clustering.(2)A new Carleman's estimation formula based on(l+x)1/x approximation.The bounds of(1+x)1/x2 is taken as the main tool for improving Carleman's inequality.There are two key issues for an inequality,one is to find the bounds,and the other is to prove the bounds.This paper takes(1+x)1/x as an example,and presents a Pade-approximant-based method for finding the two-sided bounds of(l+x)1/x with much better approximation effect,and also provides a new way for proving the bounds,which can be applied for more other inequalities.Numerical results show that it achieves a much smaller approximation error than those of prevailing methods.The results can also be applied to the Logarithmic Shadow Maps algorithm(LogSM).(3)Quadratic surface approximation based method for point projection problems of a Bezier surface.The point projection problem has wide applications in computer graphics and geometric modeling.Prevailing subdivision-based methods can ensure to achieve the global minimum distance,but seems to be time-consuming by comparing with Newton's method.This paper presents a quadratic surface approximation based method for the point projection problem of Bezier surfaces.Firstly,it computes the control net of the distance function,and achieves several local minimum control point(LMCP).Secondly,it approximates the local area nearby each LMCP by using quadratic surfaces,and obtains a better initial value for numerical iterative methods.Thirdly,it applies the Newton's method for the proper solutions.Comparing with prevailing subdivision-based methods,it saves a lot of computation time of clipping and achieves a much better computational efficiency.