CAGD中曲线插值若干问题的研究
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摘要
本文主要对CAGD中的三类插值曲线,即Ferguson曲线、多结点样条插值曲线和代数插值曲线进行了研究,提出了两类新的Ferguson曲线--HP-Ferguson曲线和H-Ferguson曲线,两类新的多结点样条--三角多项式多结点样条和双曲多项式多结点样条,三类带参数的多结点样条。以及基于几何约束的代数插值曲线在构造给定多边形的切线系中的应用。
首先介绍了两类数学描述曲线--逼近型曲线和插值型曲线,后者包括多项式插值曲线、分段三次埃尔米特插值曲线、三次参数样条曲线、多结点样条函数插值曲线、代数曲线插值、奇异混合插值、圆弧样条插值等。并以插值曲线的特点即曲线点点通过为基础,分别在三角函数空间和双曲函数空间中构造类似传统Ferguson和多结点样条插值曲线特性的广义Ferguson曲线和广义多结点样条插值曲线。它们继承了Ferguson曲线和多结点样条曲线的特点,曲线表示简单、直观。此外它们还具有三角函数和双曲函数的优点。特别地,由于两类广义曲线引入了参数α,所以曲线还可以将α用以调整曲线形状。最后由于这两类曲线由三角函数和双曲函数构成,所以它们较易转化为有理多项式曲线。从而融入到现有的几何造型系统中。
在近年来研究形状参数的变化对样条曲线形状的影响的启示下,本文进一步构造了三类带形状参数的多结点样条插值曲线,研究了参数λ的改变对曲线形状的影响。并随后给出了确定基于一阶、二阶导矢约束条件下的多结点样条插值曲线的方法。
最后介绍了基于几何约束的三次代数插值曲线,并给出了其在构造给定多边形切线系中的应用,最后给出了实例,并画出图形。
This paper summaries the researches on interpolantion curves of three types which includes Ferguson curves many-knot splines and algebraic curves.Putting forward to two new Ferguson curves--the HP-Ferguson curve and H-Ferguson curve ,two types of many-knot splines-triangle polynomial many-kont and hyperbolic polynomial many-knot, three types take parameter of many-knot splines, paper gives algebraic curves interpolation with geometric constraints finally, and giving an application which constructs the polygon tangent fasten.
First, this paper introduces two types of shape mathematics description curve—approach curve and interpolation curve,the latter includes polynomial interpolants, subsection three hermite interpolants, three parameter curves, many-knot splines, algebra curves、singular blending splines, arc splines etc...By analyzing the characters of interpolation curves namely the curves should to pass every node, respectively construct in triangle function space and hyperbolic function space, similar characteristic of traditional Ferguson and many-kont interpolation curves. Generalized Ferguson and many-kont splines inherited characteristics of Ferguson curve and many-knot splines, the curve means in brief and keeps a view. In addition, they have the advantage of triangle function and hyperbolic function. As taking parameter in two types of generalized curves in particular, so the curve can also adjust curve shape. Finally, because of constitutes to from triangle function and hyperbolic function, so they convert into the rational polynomial curves more easily. Integrate several existing shape system.
With the illuminating of the variety upon the curves shape follow with shape parameter in recent years, this paper constructed further three types many-kont interpolation curve which take shape parameter, studying the influence of the change upon the parameter.later, giving the method of mant-knot splines which is according to a rank, two ranks differentia-vector constraints.
At last, introducing cubic algebraic curves interpolation with geometric constrains, and giving an application which constructs the polygon tangent fasten ,,give solid example, and draw a sketch.
首先介绍了两类数学描述曲线--逼近型曲线和插值型曲线,后者包括多项式插值曲线、分段三次埃尔米特插值曲线、三次参数样条曲线、多结点样条函数插值曲线、代数曲线插值、奇异混合插值、圆弧样条插值等。并以插值曲线的特点即曲线点点通过为基础,分别在三角函数空间和双曲函数空间中构造类似传统Ferguson和多结点样条插值曲线特性的广义Ferguson曲线和广义多结点样条插值曲线。它们继承了Ferguson曲线和多结点样条曲线的特点,曲线表示简单、直观。此外它们还具有三角函数和双曲函数的优点。特别地,由于两类广义曲线引入了参数α,所以曲线还可以将α用以调整曲线形状。最后由于这两类曲线由三角函数和双曲函数构成,所以它们较易转化为有理多项式曲线。从而融入到现有的几何造型系统中。
在近年来研究形状参数的变化对样条曲线形状的影响的启示下,本文进一步构造了三类带形状参数的多结点样条插值曲线,研究了参数λ的改变对曲线形状的影响。并随后给出了确定基于一阶、二阶导矢约束条件下的多结点样条插值曲线的方法。
最后介绍了基于几何约束的三次代数插值曲线,并给出了其在构造给定多边形切线系中的应用,最后给出了实例,并画出图形。
This paper summaries the researches on interpolantion curves of three types which includes Ferguson curves many-knot splines and algebraic curves.Putting forward to two new Ferguson curves--the HP-Ferguson curve and H-Ferguson curve ,two types of many-knot splines-triangle polynomial many-kont and hyperbolic polynomial many-knot, three types take parameter of many-knot splines, paper gives algebraic curves interpolation with geometric constraints finally, and giving an application which constructs the polygon tangent fasten.
First, this paper introduces two types of shape mathematics description curve—approach curve and interpolation curve,the latter includes polynomial interpolants, subsection three hermite interpolants, three parameter curves, many-knot splines, algebra curves、singular blending splines, arc splines etc...By analyzing the characters of interpolation curves namely the curves should to pass every node, respectively construct in triangle function space and hyperbolic function space, similar characteristic of traditional Ferguson and many-kont interpolation curves. Generalized Ferguson and many-kont splines inherited characteristics of Ferguson curve and many-knot splines, the curve means in brief and keeps a view. In addition, they have the advantage of triangle function and hyperbolic function. As taking parameter in two types of generalized curves in particular, so the curve can also adjust curve shape. Finally, because of constitutes to from triangle function and hyperbolic function, so they convert into the rational polynomial curves more easily. Integrate several existing shape system.
With the illuminating of the variety upon the curves shape follow with shape parameter in recent years, this paper constructed further three types many-kont interpolation curve which take shape parameter, studying the influence of the change upon the parameter.later, giving the method of mant-knot splines which is according to a rank, two ranks differentia-vector constraints.
At last, introducing cubic algebraic curves interpolation with geometric constrains, and giving an application which constructs the polygon tangent fasten ,,give solid example, and draw a sketch.
引文
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