【灌水】Gn的问题[数学与几何看来要打架]

  Continuity describes the behavior of curves and surfaces at their segment boundaries. The two types of continuity usually dealt with in Unigraphics NX are mathematical continuity, denoted Cn, where n is some integer, and geometric continuity, denoted Gn. Within Unigraphics NX these can be loosely defined.
  
Gn indicates the true degree of continuity between two geometric objects. For example, G0 means the two objects are connected, or are position continuous; G1 means they are smoothly connected up to one differentiation, or are tangency continuous. G2 means they are smoothly connected by up to two differentiations, or are curvature continuous; G3 means they are smoothly connected by up to three differentiations, etc. Gn continuities are representation (parameterization) independent. The curvature combs shown in the figure below illustrate these differences.
  
Cn indicates the degree of continuity between two segments of a b-curve or a b-surface in the NURB representation. Generically, C0 means the two segments are G0 connected. C1 means they are G1 connected; etc. But, C0 does not mean the two segments are just G0 connected -- they could actually be G1 or G2 connected, and so on.
  
The key point is that Gn is for real physical continuity, while Cn is one mathematical representation of it, which may not be faithful. Since NURB is an industry standard for freeform geometry, Unigraphics NX uses it. But we always try to have Cn represent the same degree of continuity as Gn, to avoid cases where a curve is G1, but has C0 junction, etc.
  
Quoting from the ICAD Surface Designer Reference manual: "C0 continuity implies that a common point exists between two adjacent segments (i.e., the segments are touching). C1 implies that there is a common point and the first derivatives of the polynomials (i.e., the tangent vectors) are the same. C2 implies that the first and second derivatives are the same. Geometric continuity is less strict than mathematical continuity. G0 and C0 are equivalent, that is, the segments are positionally continuous. G1 implies that the tangent vectors are equal in direction, but not magnitude. G2 implies the curvature is the same, but the second derivatives are not."
  

    有耐心的看看，我已经看过一遍了。
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[效果吗？看过就知道，不看绝对不知道]

::l不太理解，怎么办？::?::?::?

翻譯一下：
   
连续性
  
  连续性描述分段边界处的曲线与曲面的行为。在 Unigraphics 中通常使用的两种连续性是数学连续性（用Cn 表示，其中 n 是某个整数）与几何连续性（用Gn 表示）。在 Unigraphics 中，可松散定义这两种连续性。
  
Gn 表示两个几何对象间的实际连续程度。例如，G0 意味着两个对象相连或两个对象的位置是连续的；G1 意味着两个对象光顺连接，一阶微分连续，或者是相切连续的。G2 意味着两个对象光顺连接，二阶微分连续，或者两个对象的曲率是连续的； G3 意味着两个对象光顺连接，三阶微分连续等。Gn 的连续性是独立于表示（参数化）的。下图显示的曲率梳状线图示了这些差异。
  
Cn 表示 NURB 表达中的 b 曲线或 b 曲面的两个段间的连续程度。一般说来，C0 意味着两个段是 G0 连接的。C1 意味着两个段是 G1 连接的等等。但是，C0 并不意味着两个段只是 G0 连接的 - 实际上它们可以是 G1 或 G2 等连接的。
  
关键的一点是 Gn 用于表示实际物理连续性，而 Cn 是实际物理连续性的数学表达，这种用法并不可靠。因为 NURB 是自由曲面几何的行业标准，所以，Unigraphics 使用它。但是，我们总是试图让 Cn 与 Gn 表示相同的连续程度，以避免出现曲线是 G1，而有 C0 连接点的情况。
  
“ICAD 曲面设计人员参考”手册中这样描述：“C0 连续性意味着两个相邻段间存在一个公共点（即两个段相连）。C1 意味着有一个公共点，并且多项式的一阶导数（即切向矢量）是相同的。C2 意味着一阶导数和二阶导数都相同。几何连续性没有数学连续性严格。G0 和 C0 的意思相同，即段在位置上连续。G1 意味着切向矢量的方向相同，但模量不同。G2 意味着曲率相同，但二阶导数不同。”
  
  

非常感谢翻译！！DING！！！！！

看不到哪里打架阿::?::?

看的晕……
太理论了！

喝喝，，头有点晕，，不过要顶一下！！！

IceFai wrote:
看不到哪里打架阿::?::?

这里吗？
“C0 并不意味着两个段只是 G0 连接的 - 实际上它们可以是 G1 或 G2 等连接的”

classic!!!