The basic operation in the development of subdivision curves is the refinement procedure. In the cubic case, this can be written as
In this case, we can also classify points of the refinement as vertex points and edge points, and this enables us to look at the problem from another point of view. Here we examine what happens to a particular control point (vertex point) under this refinement procedure. Examining this closely, we can develop an alternate matrix equation that can also be used to represent the refinement procedure.
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Refinement at a Vertex Point
The general idea in this development is to focus on a single vertex point of the control polygon and the two points immediately adjacent to this vertex. Call this vertex and the two points and - as in the following figure.
In the refinement procedure, we note that vertex points and edge points alternate in the refined control polygon. We will use these three points to create a new vertex point and two new edge points and .
We can write this in vector form (we note that the matrices are
applied to vectors of points, and therefore the operations must be
Writing this in vector form
So How Do You Do Refinement In General?
Given a control polygon , we utilize the vectors
It is possible to write the refinement process for cubic subdivision curves in a matrix form which focuses on the action of the refinement on a single control point. In this case, we can calculate new vertex points and edges points just by applying the refinement matrix to the vertex-edge-point vector.
We note that this procedure does take more processor time than the refinement based upon binary subdivision. However, it will enable us to analyze the refinement operation further and generate a procedure to directly calculate a point on the curve without resulting to refinement.