曲面

And when adjusting parameter t, a family of subdivision surfaces can be constructed.

文中对这族细分曲面的性质进行了较详尽的分析，并证明了在参数t不等于零时这族曲面在非奇异点处至少具有C~1连续。

An explicit mathematical representation of the exact boundary of a ball Bézier surface was given by recurring to the envelope theory of the family of bivariate surfaces in the classic differential geometry, as well as the spherical coordinates and the Cramer's rule.

根据经典微分几何中双参数曲面族的包络原理，运用球面参数坐标和Cramer法则，给出了球域Bézier曲面边界的精确数学显式表达式。

An instance was given by means of the numerical solution for the first boundary value inverse problem of Laplace's equation.

将直匀流绕物面作二维无旋不可压缩定常流动时的流函数表达为曲面后，利用流函数曲面控制边界与流曲线的对应关系，采用拉普拉斯方程第一边值逆问题的数值解法，提出了来流为直匀流的流曲线造型方法并给出了实例。

More specifically, Gauss studied the geometry of surfaces based on the first fundamental form (also called "line element") of surfaces and generalized Euclidean geometry to "curved geometry" on surfaces.

局部微分几何的一个里程碑是 Gauss关于曲面的理论，他建立了基于曲面第一基本形式的几何，并把欧几里得几何推广到曲面上"弯曲"的几何。

The merit of the invention are: 1 adopting free curved surface imaging method , three part reflection type Zernike polynomial free curved surface forms short focal distance bigger, relative apertures bigger fields of view angle imaging , rectifying optical aberration much better and its structure is simple; 2 adopting the reflection structure without optical lens, has removed the effect of various chromatic aberration , guarantee the imaging articulation; 3 adopt the design of short focal distance big fields of view angle, focal distance 8 - 15 mm, field of view angles 100 -135 degree, having 120 times multiplications under 250 mm imaging distance, having ultra-thin characteristic property; 4 adopting big relative apertures design ,the relative aperture is 2.0-2.4, having improved the illuminance of flat surface; 5 adopting image space telecentric imaging method , ensures that the image surface irradiance is homogeneous.

本发明的优点是：1采用自由曲面成像方法，三片反射式Zernike多项式自由曲面形成短焦距大相对孔径大视场角成像，更好地校正像差，结构简单；2采用无透镜形式的反射式结构，消除了各种色差影响，保证成像清晰度；3采用短焦距大视场角设计，焦距8～15mm，具有100～135度的视场角，在250mm成像距离下有120倍的放大倍数，具有超薄特性；4采用大相对孔径设计，相对孔径为2.0～2.4，提高像平面上的光照度；5采用像方远心成像方法，保证像面辐照度均匀。

In chapter 5, the forked pipe blending surfaces modeling based on the envelope of the rolling ball is studied.

在这一章末，讨论了负高斯曲率的Bezier曲面在房顶曲面设计中的应用。

Meanwhile it proves the local property of four-point subdivision scheme, which provides the theory foundation for 3D garment simulation and interactive design using four-point subdivision scheme.

同时，本文还推导出四点细分曲面的局部修改性，为采用四点细分曲面进行三维服装模拟和交互式设计提供了理论基础。

Since the higher quality of free-form surfaces are required, it is very important to measure the free-from surface accurately and quickly.

由于对自由曲面质量要求的不断提高，实现自由曲面的快速、精确测量已成为一个重要的研究课题。

Finally some theories and methods on free-form surface and blending surface feature extracting are discussed.

此外，还研究了自由曲面和过渡曲面特征提取的一些理论和方法。

The triangle mesh model of surface was extracted by constructin g an implicit surface function.

首先建立插值散乱数据点的隐式曲面方程，然后使用步进立方体算法提取出曲面的三角网格模型。

Breath, muscle contraction of the buttocks; arch body, as far as possible to hold his head, right leg straight towards the ceiling (peg-leg knee in order to avoid muscle tension).

呼气，收缩臀部肌肉；拱起身体，尽量抬起头来，右腿伸直朝向天花板（膝微屈，以避免肌肉紧张）。

The cost of moving grain food products was unchanged from May, but year over year are up 8%.

粮食产品的运输费用与5月份相比没有变化，但却比去年同期高8％。