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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.