Rune wrote:
> 
> Le Forgeron wrote:
> > Ellipsoid is only a linear transformation of a
> > sphere, so it won't help when the normals do not meet.
> 
> Does that logic apply?
> 
> Three normals of a sphere will always meet in the same point. But three
> normals of a ellipsoid will not always meet in the same point. I'm not
> saying that an ellipsoid is a solution, but just questioning this
> particular argument.
> 

Maybe you're right, and for some cases you can find an ellipsoid 
which could accomodate the three normals (even if I have no idea of
how to find the ellipsoid parameters; using a unit sphere is easier
for the math). 
But a linear transform cannot transform a positive curvature into
a negative one. So the ellipsoid won't solve all the cases.

Given also the troubling case where two normals are parallel (which
cannot be solve on someting like a sphere or even an ellipsoid),
I wonder if the curved triangle shouldn't be on a torus instead (*).

At least for the two parallel normals case, it seems that two vertices 
would be on the very top circle of the torus with the third vertex somewhere
so that its normals is ok. Currently, I can only figure the math to 
position the first vertex, the circle where the second vertex is and 
where the third vertex might be.
Maybe using also the true normal of the initial triangle might help to
limit the position of the second and third vertices. 

Once the triangle on the torus is defined, we still go back to the classical
transformation of three points from one space to another, nothing really difficult!
But maybe the torus is not even the right solution.

(*): The more I look at the problem, the more it make me think of an analogy with 
conic curves (the 6 classical conic curves in 2D can all be view as the intersection
of a plane and an infinite double cone: ellipse, parabol, hyperbol, point, single and
double generating lines), which would means that to solve the really curved triangle,
you have first to find out which 3D objects should be used according to the
parameters...
Looks like an intersection of hyperplane with an hypercone, where the position of the
hyperplane
is made according to the parameter, isn't it ?
Only problem is that I can imagine in 3D, but 4D is a bit too much (excepted when it's
3D+Time,
which is not the case here)

-- 
Non Sine Numine
http://grimbert.cjb.net/
Etiquette is for those with no breeding;
fashion for those with no taste.

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