Continuity
 List of: Discussion Topic Subjects: Mathematics Contents: Fundamental Concepts
In a general sense, continuity describes how two items come together. In ACIS, these items may be two curves that meet in some way, two portions of the same curve, etc. (In the latter case, one is usually describing the smoothness of a curve, which is a global property, in terms of a local property.)

In ACIS, two types of continuity are generally discussed: Cn and Gn, where n refers to the nth derivative. Cn continuity refers to continuity of the nth derivatives of the equations underlying the entities. This means that the magnitude and direction of the nth derivative must be continuous. Gn continuity refers to continuity of geometric, or parameterization-independent, properties, which means that only the direction of the nth derivative must be continuous. The difference between the two types of continuity is that Gn allows the parameterization to be changed to achieve desired continuity. By manipulating the parameterization of the curve or surface, one can change the magnitude of vectors.

For example, if two entities meet with C0 continuity, their zeroth derivatives are the same at their intersection. (In the case of C0 continuity, it may simply be said that the entities "are continuous.") At all points along the intersection, the position of the entities are the same. G0 continuity also means that the zeroth derivatives are the same at their intersection, but changing the parameterization of one of the entities doesn't affect its position.

The following list describes what is meant if two items (e.g., curves) meet with the specified continuity.

C1 continuity Means that the first derivatives, or tangents, are identical (in addition to C0 continuity). The tangents of curves and surfaces are vectors, so both the magnitude and direction of the tangent vectors must be identical.

G1 continuity Means that just the direction of the tangent vectors are identical, which is not as stringent as C1. By changing the parameterization of a curve or surface, one can affect the magnitude of the tangent vectors without affecting the direction.

C2 continuity Means that the second derivatives agree (in addition to C1 continuity). Because curvature is a function of the first and second derivatives, one often says that the curvature is continuous if entities are C2.

G2 continuity Means that just the direction of the second derivatives are identical, which is not as stringent as C2. One can change the parameterization of one of the entities to get the geometric curvatures (independent of parameterization) to agree.

The derivative level, n, to which an object is continuous refers to its degree of continuity. If a given object is continuous at the nth derivative, it is said to have nth degree of continuity (or degree of continuity n). To state that a given curve has a particular degree of continuity means that for all points on the curve's interior, the continuity is at least of that degree. The same holds for surfaces.
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