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#9 Circular/elliptical segments in curves
Antoon Pardon on sketch-list:
Allow a circle segment in a curve. The way I see this
done is by giving a control point as with a bezier
curve. But the control point would only be used to
control the tangent in the point, together with the two
point of the segment this would specify the circle
segment.
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Antoon Pardon provided some more thoughts on sketch-list
(posting
http://www.geocrawler.com/lists/3/SourceForge/5016/0/5781166/
):
Well I have considered three approaches.
The first and second try to specify an ellips by giving two
nodes and one controlpoint. The controlpoint specifies the
tangents in the nodes. Now one way to make this a complete
specification of an elliptical arc is by interpreting the
two nodes and the controlpoint as the result of a
transformation applied to the points (1,0), (0,1) and (1,1).
The elliptical arc specified would be the result of the
first quarter of a circle applied to the same
transformation. The disadvantage of this approach is that it
only allows circular arcs of 90 degrees and similar
limitations to elliptical arcs.
A second approach would by starting from a circle segment.
The start and endpoint are the nodes and the tangents in the
start and endpoint cross in the controlpoint. Transforming
this setup would transform the circle arc in an elliptical
arc. However there is more than one cobination of circle arc
+ transformation that leads to the same result of Nodes and
controlpoint. Unfortunatly they don't result in the same
elliptical arcs. Of course we could specify a specific
method and in this way come to a well specified result. The
problem however is that whichever method you choose, you
allways have the same problem. The curve specified by the
transformed points is not the transformed curve specified by
the points.
The third approach I considerd was by introducing a second
controlpoint. This second controlpoint would specify the
middlepoint of the ellips. (One could consider the case of
it specifying a focus). Now the idea is to combine the
previous approach with the extra controlpoint we have. As
far as I can see, this would work under affine
transformations. The only drawback is that not all choices
of nodes and controlpoints will lead to an elliptical arc.