OPEN CASCADE Curve Continuity

2015/11/17 23:41

OPEN CASCADE Curve Continuity

eryar@163.com

Abstract. 设计一条复杂曲线时，出于设计和制造上的考虑，常常通过多段曲线组合而成，这就需要解决曲线段之间如何实现光滑连接的问题。评价曲线间连接的光滑度的度量有两种：参数连接性和几何连续性。本文对这两种连续性分别进行介绍。

Key Words. Curve Continuity, Geometric Continuity, 参数连续性、几何连续性

1.Introduction

2.Parametric Continuity

3.Geometric Continuity

4.Curve Continuity

Φ(t)在[0，2]上表示一条连接V0，V1的直线段，但却有

Φ(t)明明是一条直线，却非C1连续，说明用参数连续性描述光滑性是不恰当的。

Figure 4.1 两条曲线拼接的连续性

β为任意常数。当α＝1，β＝0时，G2连续就成为了C2连续。至此可以看到，C1连续保证G1连续，C2连续保证G2连续，但反过来不行。也就是说Cn连续的条件比Gn连续的条件要苛刻。

OPEN CASCADE中关于曲线是连续性的定义使用了GeomAbs_Shape枚举定义：

//! Provides information about the continuity of a curve:
//! -   C0: only geometric continuity.
//! -   G1: for each point on the curve, the tangent vectors
//! "on the right" and "on the left" are collinear with the same orientation.
//! -   C1: continuity of the first derivative. The "C1" curve is
//! also "G1" but, in addition, the tangent vectors " on the
//! right" and "on the left" are equal.
//! -   G2: for each point on the curve, the normalized
//! normal vectors "on the right" and "on the left" are equal.
//! -   C2: continuity of the second derivative.
//! -   C3: continuity of the third derivative.
//! -   CN: continuity of the N-th derivative, whatever is the
//! value given for N (infinite order of continuity).
//! Also provides information about the continuity of a surface:
//! -   C0: only geometric continuity.
//! -   C1: continuity of the first derivatives; any
//! isoparametric (in U or V) of a surface "C1" is also "C1".
//! -   G2: for BSpline curves only; "on the right" and "on the
//! left" of a knot the computation of the "main curvature
//! radii" and the "main directions" (when they exist) gives the same result.
//! -   C2: continuity of the second derivative.
//! -   C3: continuity of the third derivative.
//! -   CN: continuity of any N-th derivative, whatever is the
//! value given for N (infinite order of continuity).
//! We may also say that a surface is "Ci" in u, and "Cj" in v
//! to indicate the continuity of its derivatives up to the order
//! i in the u parametric direction, and j in the v parametric direction.
enum GeomAbs_Shape
{
GeomAbs_C0,
GeomAbs_G1,
GeomAbs_C1,
GeomAbs_G2,
GeomAbs_C2,
GeomAbs_C3,
GeomAbs_CN
};

5.Conclusion

6.References

1. 莫蓉. 常智勇. 计算机辅助几何造型技术. 科学出版社. 2009

2. 王仁宏. 李崇君. 朱春钢. 计算几何教程. 科学出版社. 2008

3. 孙家广等. 计算机图形学. 清华大学出版社. 2000

4. 朱心雄. 自由曲线曲面造型技术. 科学出版社. 2008

5. Shing Liu. OPENCASCADE Curve Length Calculation.  http://www.cppblog.com/eryar/archive/2014/08/25/208127.html

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