"lelama" <nomail@nomail> wrote:
> > I've posted these to see if there is enough interest for me to either post code,
> > bugs and all, or spend (quite a bit of) time cleaning it up and uploading to
> > objects.
> >
>
> Thanks Jim, this is an interesting idea.
>
> I made some trials on the same idea although my computations were the other way
> round : from the three spheres, let say in the horizontal plane, I first
> computed the 2 tangent planes to the three spheres, the one above the spheres
> and the one below.

I accept your argument. Three spheres will have a pair of tangent planes and if
you join two of the tangent points, you have a line with two identical normals
at the ends which must be a generator of the tangent cone. The fat triangle
therefore has to be C_1 and the apparent discontinuity is all to do with
discontinuity of curvature.

Coming from the tangent cones direction, I didn't see why the two tangent points
had to be on a single generator of the tangent cone and assumed they weren't.
Checking in an actual example:

Nor21    = <-0.59960801,-0.05058799,0.79869336>
Nor22    = <-0.59960801,-0.05058799,0.79869336>
Nor23    = <-0.59960801,-0.05058799,0.79869336>

that is, three identical normals to 8 dp. So, thanks, I now realise the smooth
triangles are not needed - ordinary triangles or a pair of transformed prism
objects will do (a prism would help with CSG intersections and differences).

Simplifying the construction of a single fat triangle should have an effect on
the speed of parsing - which would be a help since it is slooow.

>From the 6 tangency points, we get 6 lines. Grouping these
> lines by pair, we get 3 cones. More specifically, in the middlee of each pair is
> a line through the centers of 2 spheres, and the corresponding lathe object
> around the middle line is the cone to be added.
>
> This construction shows if I am not wrong that the cone is tangent to the plane,
> so I did not understand your C1-problems in your first message.
>
> My goal was much more simple than what you did. I just wanted to draw rounded
> boxes with different radius along the edges.

I've added an image with the triangles in a different colour. I initially
expected the edges to be subtly curved, which they are not. Printing out the
three normals is a confirmations.

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