Pasting

The pasting process itself works as follows:

1. The domain of each surface is embedded in the space of the surface.
2. We then find a mapping of the feature domain into the base domain.
3. Now, for each control point of the feature domain, we construct a coordinate frame, F1, for this control point in the feature domain, using the Greville Abscissa as the origin and the coordinate axes as basis vectors. Express the feature control point relative to this coordinate frame.
4. We compose the feature domain/frame with the base surface to construct a new coordinate frame, F2, on the base surface.
5. Extract the feature's control points relative to F1 and use the to weight the frame F2. This give the location of the pasted feature control point. Once we have performed this pasting with all the control points, we can evaluate the feature as we would any other tensor product B-spline.

   Note that the boundary of the feature will only approximately lie on the base. However, by performing knot insertion, we can make this approximation good to any tolerance.

   

   In general, we will want an arbitrary hierarchy of domains. One domain, however, must be the root of all other domains (i.e., the other domains must be subdomains within this root domain). Thus, we can represent the domain hierarchy as a directed acyclic graph with a single root.

   Up one level.