Study On The Problem Of Shape Preserving Trigonometric Cubic Spline Based On Function Values

The construction method and shape preserving of curves and surfaces are the key problems in Computer Aided Geometric Design(CAGD).One of the most powerful tools is spline interpolation.In recent years,spline interpolation with parameters received wide attention.The curves and surfaces can be modified by suitable parameters under the condition that the interpolating data are not changed.Then the shape control of curves and surfaces is achieved.Firstly,this paper introduces the research background and development history of CAGD.The development of trigonometric spline curves and surface with shape parameters are given.It includes the Bézier curves and surfaces with shape parameters,B spline curves and surfaces with shape parameters,rational spline curves and surfaces with shape parameters.Then in the second part we construct a cubic trigonometric spline interpolation with parameters.It defines three parameters in each subinterval.The two parameters have been constrained to preserve the positivity,constraint and monotonicity.Another free parameter is to adjust the curves' shape.Then,some numerical examples are given to verify the positivity,constraint and monotonicity of the curves.In the third part,the developed cubic trigonometric spline is extended to bivariate cubic trigonometric spline surface,and then the positivity and monotonicity preserving conditions of bi-cubic trigonometric spline surface are given.When the boundary curves maintain positivity and monotonicity,the interpolation surface is also positive and monotone.That is,by adjusting the positivity and monotonicity of boundary curves,the positivity and monotonicity of the surface are achieved.The last part is the conclusion of this paper.