曲面族

After obtaining a family of blending curves and surfaces, we construct a new energy smoothing equation for determining the parameters to get the desirable blending curve and surface.

另外，在得到的混合曲线曲面族中构造出了新的更优能量光顺方程，通过设置参数可得到合适的混合曲线曲面。

In this manner, a developable surface was designed using control planes with Bézier and B-spline basis functions.

基于将可展曲面作为平面单参数族包络面的思想，给出在Bézier和B样条基下可展曲面的参数表示形式。

First, this paper, in the field of intrinsic geometry, studies the geometric problems on garment design, as well as applies the frame and semi-geodesic coordinates to prove the fundamental theorem of being a developable surface.

文中首先在内在几何学的层次上，研究了服装设计所涉及的几何学问题，应用标架与半测地坐标方法证明了曲面成为可展面的基本定理，研究了可展面的分类及其性质，考虑到服装三维几何造型的需要，证明了组合式可展面各组成片相切连接条件的命题，作为构造可展面的理论依据，证明了单参数平面族的包络面必为可展面的命题，在此基础上发展出服装几何造型的"刮大白"方法以及相关的三种构造可展面的解析方法。3D→2D的变换是三维服装CAD的重要内容之一，其几何学实质是曲面的定长映射，文中总结了定长映射即可展面在平面上展开的基本准则，在这一准则的指导下，结合服装设计与相关领域的要求，讨论了可展面在平面上展开的解析方法与数字方法，上述内容确立了服装设计几何学的基本框架。

We proved that the necessary and sufficient condition that the developable surfaces touch the second focal surface along the orthogonal trajectories was that the base surface S was a developable surface.

证明了线汇中的两系可展曲面交第二叶焦曲面于正交轨线族的充要条件是原曲面是可展曲面，并给出了两叶焦曲面在同一条光线上两焦点相互对应下，保持渐近曲线相互对应的充要条件。

It shows that some of the geometric construction algorithms that exist for curves can be also generalized to the design of developable surfaces.

基于将可展曲面作为平面单参数族包络面的思想，给出在B啨zier和B样条基下可展曲面的参数表示形式。

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。

We prove that the two families of torsal surfaces of $T$ touch the second focal surface $S_2$ along the net of orthogonal parametric curves if and only if $S_1$ is developable.

证明了$T$的两个可展曲面族沿着第二个焦曲面$S_2$的正交曲线网相交的充要条件是$S_1$是可展曲面。

By anchoring an imaginary spring between each data point and deformable curve or surface, the fairing model is established. Then FEM is used to solve the deformation problem of the model caused by imaginary springs and a bicubic B-spline surface patch is taken as one element. Lastly the B-spline active surface model is applied to surface reconstruction from layer data and the problem resulted from fitting for uneven cross-sectional curves can be solved.

通过在数据点与模型之间连接虚拟弹簧，建立光顺模型，并以双三次B样条曲面片作为单元，运用有限元法对模型在虚拟弹簧作用下的变形问题进行求解，得到光顺后的B样条曲面；并在此基础上，提出了一种新的基于断层数据的表面重建方法，可很好地解决不均匀截面族的曲面生成问题。

We consider the rectilinear congruence $T$ generated by the tangents to a one parameter family of geodesics on a space-like surface $S_1$ in the Minkowski $3$-space $E_1^3$, having $S_1$ as one of its focal surfaces.

研究了由三维Minkowski空间$E^3_1$中一个类空曲面$S_1$上一个单参数测地曲线族的切线所构成的直线汇$T$，它以$S_1$为一个焦曲面。

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％。