Thank you for your interest in the solution I proposed.

adrianv avm4@cornell.edu wrote:

Yes, the border of the standard 5th degree tripatch is indeed 4th degree
Bezier curve like the ones we considered for 2D cases with curvature
continuity. Its degree was elevated to 5 in order to have more freedom in
the internal control points of the patch. The control points of the 3
border curves of the standard 5th degree tripatch are:

[[1, 1, 0], [1, 0.6, 0], [1, 0, 0], [1, 0, 0.6], [1, 0, 1]]

[[0, 1, 1], [0, 1, 0.6], [0, 1, 0], [0.6, 1, 0], [1, 1, 0]]

[[1, 0, 1], [0.6, 0, 1], [0, 0, 1], [0, 0.6, 1], [0, 1, 1]]

They correspond to 4th degree curves with a form factor of r0=0.4.
Another possible form factor appeared in the development of the tripatch.
Its effect is restricted to the interior of the patch and do not affect its
border. That form factor were arbitrated to be 1 in the standard 5th degree
tripatch I proposed before.

It seems to me that to compare the standard tripatch with the degenerated
rectangular one, this last one should be generated with the same form
factor 0.4. Otherwise, the corner border curves (and the edge roundover)
may be very different.

adrianv avm4@cornell.edu wrote:

What do you mean here by "right direction"?

If you give me the form factor value you are using in your degenerate
rectangular roundover , I could generate a standard 5th degree tripatch
that meets the same form factor. Besides the form factor itself, it would
be nice to have the patch border control points to check against mine patch.

Ronaldo wrote

I agree, hence the inconclusive nature of the test.

However, I am definitely bothered by the funky highlights.  In the pictures
the tripatch is on the left, degenerate rectangle in the middle, and
circular roundover on the right.  The highlights look odd, which in an of
itself is a negative characteristic of the printed model.  It's probably the
most noticeable characteristic of the curves.  It seems like the tripatch
has some sort of bulge compared to the other ones.

http://forum.openscad.org/file/t2477/hilights.jpg
http://forum.openscad.org/file/t2477/highlights2.jpg

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Ronaldo wrote

To use more precise language:  There exists a direction from which the
highlight line on the model appears to have a corner (highlight has a
discontinuity in its derivative).  This is visible in the photo I posted in
my other message.

I generated my patch like this:

Q = cornerPatchCP([0,0,0],1.5, 0.3);

using the cornerPatchCP that you previously posted.  I was comparing this
patch to the patch you originally posted, so the length of d disagrees, I
think.  I can change the patch, but for print test a larger curve is better.

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adrianv avm4@cornell.edu wrote:

I understand it now. In fact, the images suggest something odd in the
border of the tripatch. See bellow.

I generated my patch like this:

The standard patch has a unitary "radius". To get a bigger roundover to
print, just scale it up.
The value 1.5 of the second argument in cornerPatchCP correspond to scale
up the standard patch by 1.5.
To match the form factor of the border curves of  cornerPatchCP([0,0,0],1,
0.3) you may use the following standard patch CPs:

[[0, 1, 1]]

[[0, 1, 0.76], [0, 0.76, 1]]

[[0, 1, 0.28], [0, 0.7, 0.7], [0, 0.28, 1]]

[[0.28, 1, 0], [0, 0.55, 0], [0, 0, 0.55], [0.28, 0, 1]]

[[0.76, 1, 0], [0.7, 0.7, 0], [0.55, 0, 0], [0.7, 0, 0.7], [0.76, 0, 1]]

[[1, 1, 0], [1, 0.76, 0], [1, 0.28, 0], [1, 0, 0.28], [1, 0, 0.76], [1, 0,
1]]

That patch may be somewhat bulged when compared with a sphere, for
instance.

The best approximation I got of a sphere surface with a G2 tripatch has the
following CPs:

[[0, 1, 0.904], [0, 0.904, 1]]

[[0, 1, 0.352], [0, 0.88, 0.88], [0, 0.352, 1]]

[[0.352, 1, 0], [0, 0.52, 0], [0, 0, 0.52], [0.352, 0, 1]]

[[0.904, 1, 0], [0.88, 0.88, 0], [0.52, 0, 0], [0.88, 0, 0.88], [0.904, 0,
1]]

[[1, 1, 0], [1, 0.904, 0], [1, 0.352, 0], [1, 0, 0.352], [1, 0, 0.904], [1,
0, 1]]

whose border curves should match the border curves of cornerPatchCP([0,0,0],1,
0.12).

The second form factor I mentioned before does not affect the patch border
but controls the bulge of the patch. When we decrease its value, but
keeping the border curves fixed, the patch curvature varies very fast,
although continuously, near the patch border which may appear as a
discontinuity. Which means that the 3rd derivative near the border is
higher. Perhaps, that is the cause of the highlight behavior we observe in
your images. The last patch above seems to be a reasonable compromise.