Le Forgeron <jgr### [at] freefr> wrote:
> Le 06.10.2007 13:33, honnza nous fit lire :
> > Is there an easy way to join two points by two circular arcs given their
> > tangents?
> > Input:
> > A,B - endpoints
> > a,b - their tangents (oriented parallel to the arcs)
> > where A+at and B+bs are skew lines (it's easy to handle the planar case)
> > Output:
> > C,D - arc centers
> > arc normals and radii can be calculated easily from these
> > E - intersection point of both points
> > together with the arcs' common tangent vector it will be used to cut the
> > tori.
> >
> > I think (after some analysing) it has an infinite number of solutions,
> > parametrised by a real number.
> >
> >
> Do you also want continuity in E (intersection of both arcs): same
> tangent direction  ?
>
exactly.
>
> restricting analysis to 2D, I already have a puzzle with
>  A: <0,0>
>  B: <1,0>
>  a: <0,1>
>  b: <-1, 0>
>
> Where are C and D ?
>
There is an infinite number of two-arc solutions, with C=(1/4,0),D=(3/4,0)
being the most symmetric one. Thanks for pointing out.
>
> Even stranger, the basic:
> A: <0,0>
> B: <1,0>
> a: <0,1>
> b: <0,1>
>
>
> An obvious single arc (centered at <0.5,0>), but how to do that with
> 2 (short of degenerated solution like twice the same center) ?
>
the arc solution is a special case of the arc-line solution, the line
segment being zero. Having two non-degenerate arcs isn't neccesary, and
it's a common practice to allow degenerate solutions for special cases.
e.g. circle inversion sends circles to circles and three points always
define a circle, considering lines being degenerate circles.

The basic solution is to take a circle tangent to both lines at one of the
points, and joining the other point by a linear segment (a degenerate arc).
However, it fails in half of the cases (or it generates a solution that
crosses the infinity).
>
> Back in 3D, two points and 1 tangent might define a plane (excepted
> when tangent is colinear to delta of the points). Remains the
> question: is the second tangent in the same plane or not.
> If yes, a 2D solution is enough.
> If no, is there a solution ?
>
It's not in the same plane. I already consider this a planar case. By "skew"
I understand "not parallel and not intersecting".
I believe there's always a solution, even infinitude of them.
>
> --
> The superior man understands what is right;
> the inferior man understands what will sell.
> -- Confucius

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