Definitions and differences of curve smoothing metrics C0, C1, C2, G0, G1, and G2 [z]

There are two different measurements for curve smoothing:

One is the atomicity of the function curves that have been used for many years. The composite parameter curves have continuous vector at the link until the n order, this kind of smoothness is called Cn or parametric continuity );

Another type is geometric continuity. A combination of curves that meet a group of constraints different from that of Cn at the junction is known as geometric continuity with n order, or Gn for short.

 

According to the definition, parameter continuity is related to the obtained parameters. In fact, when the control vertex of the spline is given, the shape of the curve is completely determined (if it is NUBRS, you can also adjust the weight), and the smoothness of the curve connection is completely determined, which is irrelevant to the obtained parameter. At the same time, the practice shows that the parameter curve may be non-smooth, while the smooth curve may not. From experience intuition, we find that the two curve segments are connected, and they are considered smooth as long as they have the same tangent direction at the connection point. However, to measure smoothness Based on Parameter continuity, the same length of the tangent modulus must also be consistent with that of C1. Therefore,Parameter continuity is an over-limit on the smooth connection of parameter curves and is imposed manually.Parameter continuity is related to parameter selection and specific parameterization. Objective and internal geometric features of the shape, such as smoothness, are independent of parameter selection and specific parameterization.

 

Because parameter continuity cannot objectively accurately smooth the connection between parameter curves, it is replaced by visual continuity. In the automotive industry, there is a Class-A Surface concept, and the symbols are also G0, G1, and G2.

Curve:

C0 and G0 are consistent;

C1 and G1 are inconsistent. G1 indicates a common unit tangent;

G2 indicates a vector with a public curvature.

Surface:

C0 and G0 are consistent;

C1 and G1 are inconsistent. G1 indicates a public slice;

G2 indicates that the connection line has a public tangent plane and a public principal curvature.

 

Original: http://blog.sina.com.cn/s/blog_64eb03bb010148a5.html