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ABSTRACT This paper summarizes our researches on two key techniques in curve and surface modeling-the geometric approximation techniques and the geometric interpolation techniques. The former includes three types of geometric approximation techniques: Hybrid polynomial approximation to rational curves and surfaces, offset curves and surfaces approximation, interval Bézier curves and surfaces approximation. The latter includes a series of algorithms for shape blending modeling: the MSI algorithm for shape blending between space polylines or trees, the MSI algorithm for shape blending between triangular meshes, and skeleton-based shape blending algorithms using triangular decomposition and star-shape decomposition.

本文是作者对曲线曲面造型中两类核心技术几何逼近技术和几何插值技术的研究成果的总结，其中前者包括有理曲线曲面的Hybrid多项式逼近、等距曲线曲面逼近和区间曲线曲面逼近等三类几何逼近方法；后者包括用于形状混合造型的几何插值方法的一系列算法：空间多边形形状混合的MSI插值算法、三角网格形状混合的MSI插值算法、树的形状混合的MSI插值算法以及基于骨架树形状混合的三角剖分算法和星形剖分算法。

Jacobian Davis iterative method of solving equations AX = B, least squares polynomial fitting, portfolio Simpson formula for integration, with a triangular decomposition method of solving equations AX = B.

拉格朗日插值多项式拟合，牛顿插值多项式，欧拉方程解偏微分方程，使用极限微分求解导数，微分方程组的N=4龙格库塔解法，雅可比爹迭代法解方程AX=B，最小二乘多项式拟合，组合辛普生公式求解积分，用三角分解法解方程

As the cubic B-spline curves, each trigonometric polynomial curve segment was generated by four consecutive control points.

与三次B样条曲线类似。曲线的每一段由相继的4个控制顶点生成。

For equidistant knots, the trigonometric polynomial curves of first and second degrees are C^1 continuous, and those of third degree are C^3 continuous.

对于等距节点，一次和二次三角多项式曲线是C^1连续的，三次三角多项式曲线是C^3连续的。

The theoretical study shows that the continuous variability of non-operator, and operation, and or operation can be realized in mathematics, and can use a variety of forms such as mathematical index, polynomial and trigonometric. This paper introduces several and/or operations models, and the comparison by the simulation experiment shows that the model I is a more ideal model.

理论研究表明泛逻辑中泛非、泛与／或运算的连续可控性在数学上是完全可以实现的，并可以用多种数学形式如指数、多项式和三角函数等形式来实现，本文介绍的泛与／或运算的几个模型，通过仿真实验比较可以得出模型一就是一个比较理想的模型。

In signal processing of sonar and radar, we meet a class of nonlinear systems with alterable dimensions, and every equation in the system is a mixed trigonometric polynomial.

在声纳和雷达信号处理中，需要求解一类维数可变的非线性方程组，这类方程组具有混合三角多项式方程组形式。

For equidistant knots, the trigonometric polynomial spline curves were C^3 continuous.

对于等距节点，在一般情况下，曲线达到了C^3连续。

The authors used the method of trigonometric series and polynomial to solve the problem for the bending of a cantilever plate in a symmetric triangular shape under uniform load.

作者以前应用三角级数加多项式的方法解决了悬臂对称三角形板在均布载荷作用下的弯曲问题。

By analyzing the characters of Bezier, we construct trigonometric polynomial curves in the space of trigonometric functions, which assume the characters of Bezier curves.

本文主要工作：(1)以Bézier曲线的特点为基础，在三角函数空间上构造了具有Bézier曲线特性的三角函数多项式曲线，称其为TC-Bézier曲线。

The trigonometric polynomial curves retain the main superiority of the quadratic Bézier curves.

给出了二次三角多项式形式的Bézier曲线，基函数由一组带形状参数的二次三角多项式组成。

Researches reveal that students who suffer from corporal punishment tend to resort to violence in their future life.

研究表明遭受过肉体惩罚的孩子倾向于在他们今后的人生中使用暴力。

Total sugar of sarcocarp and chlorophyll of pericarp increased firstly, and then decreased afterward along with altitude; Titratable acidity, ascorbicacid and anthocyanin of pericarp were increased along with altitude; The content of protein increased firstly and decreased afterward along with altitude; Carotenoid of pericarp were decreased along with altitude.

果肉总糖和果皮叶绿素含量随海拔高度升高先升高后降低；可滴定酸、维生素C和果皮花青苷含量均随海拔升高而升高；蛋白质在1720m处最大，1420m处最小；果皮类胡萝卜素随海拔升高而降低。