Initial polygon

In order to look more closely at the neighbourhood of this point, let us choose the coordinate system20 in which the initial polygon is expressed, so that this point is at the origin. There is then no contribution to this... 

What happens when the control points of the initial polygon have values following some pattern ... 

The problem with this particular duality is that the dual of an inflexion in a curve is an asymptote, so that if the polygon has an inflexion the curve will contain pieces of hyperbola going off to infinity. If rules are put in place that something special is done when the polygon has an inflexion, however, this can give a nice shape-preserving scheme which has no inflexions in the limit curve when the initial polygon has none. 

We consider first how to implement the refinement process itself, then how to draw the curve defined for a given scheme and a given initial polygon, and then how to compute the primitive operations used within, for example, a Computer Aided Design software system. 

Artifacts What features can be seen in the limit curve which cannot be controlled by choice of the initial input polygon ... 

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