> scott <sco### [at] scottcom> wrote:
>>
>>
>> You'll need much more than the "impact curve" in order to build the
>> mesh. [...]
>>
>> Also what happens when the sharp edge / impact curve that is needed to
>> create the round is not visible in the image (because it's obscured by
>> another part of the shape or just pointing away from the camera)?
>> Attached example.
>
>
>
>
> If some part is not visible, we don't care: The proposed algorithm starts from
> the 2D image.

What about transparency with refraction? Or even without any refraction 
for that matter?
What about parts that are visible through reflection?
What if the non-visible part can have any notable effects on the visible 
parts?

>
> Let me give with some details the algorithm which computes the mesh of the
> rounded corner for clarity.

Tesselation is really touchy and often unreliable whan you are dealyng 
with complex shapes.
Remember that POV-Ray don't work with meshes outside the hight_field, 
mesh and mesh2 primitives.

>
>
> Let us represent the pixels on the 2D image as squares on a chess board. Let us
> call segment the piece of line which is at the interface of two adjacent squares
> on the board. For each square s on the board, by the hypothesis, we have the
> following parametrized data: an impact point p(s) in the 3D space, an object
> o(s) which contains the impact point p(s), the normal n(s) at the impact point.
> (Recall that I supposed that for each pixel in the rendering,  the impact
> surface and the corresponding normal was accessible).
>
>
> An sharpEdge is a segment such that the two adajacent squares s1,s2 on each side
> verify:
> - there are on two different surfaces o(s1)!=o(s2) and the user asked to smooth
> the intersection of these two surfaces
> - the distance between the two corresponding points p(s1) p(s2) in the 3d space
> is small enough ( small depends on the user input parameters).
>
> 2 sharpEdges are called connected if
> - they are connected on the chess board, ie they have in common the corner of a
> square
> - the two segments have the same objects on each side

In an union, the objects on each sides of an edge are effectively 
distinct and ultimately unrelated. This will cause some problems for 
your resolution.
You can't reliably tell whether any two surfaces that appear to be in 
contact in 2D are really contacting each other in 3D.

>
> ( If unclear, The last condition may be described precisely as follows
> - if the two edges form a right angle, the three squares s1,s2,s3 on the
> exterior of the curve have the same object o(s1)=o(s2)=o(s3)


between 2 3D surfaces CAN appear as a right angle depending on the 
direction from whitch it's seen.

> - it the two edges are along a line, and if s1 s2 are two squares adjacent to
> the edges and on the same side of the edges, then o(s1)=o(s2)
>
> If we put a set of connected edges together, we get a connected component of
> sharpEdges. This is a ray that we see on the 2D image where we want to smooth.
> After a moment of thouhgt, we see that a connected component is a set of
> sharpEdges which is not a general tree, but a simple branch.  For instance, in
> the example you propose, there are 3 connected components of straight edges :
> one component for each pair of surface.
>
> Thus now, we have a well defined branch, with 2 sides, which I call left and
> right for simplicity, and we want to produce a mesh around this branch.
> There is an object o(l) on the left of the branch and an object o(r) on the
> right. By construction, all the squares adjacent to the left side
> parametrize ray whose impact is in o(l) and same thing for the right.
>
>
> Now, we delimit the zone around the branch that we shall smooth out. We
> construct a set S(l) containing squares on the left side. The squares s in S(l):
> - are connected to the left side of the branch
> - are  sufficiently closed to the curve: ie there is s' adjacent to the curve
> with p(s)-p(s') small enough
> - have impact in the required surface ie o(s)=o(l)
>
> The boundary of S(l) is a union of segments which is closed curve: this is a
> deformed circles. More
> precisely, it is a union of 2 pieces: the sharp edges that we are
> trying to smooth out, and a residual piece which I call the left boundary
> B(l).
>
> Similarly, there is a right boundary B(r). THe union of B(l) and B(r) is a
> closed curve around the connected component that I call the total boundary B.
>
> The boundary B is a union of segments, and it is a deformed circle. We want to
> smooth the zone corresponding to the squares inside B,
>
> If a square s is in the circonference of the deformed circle, ie if it
> adjacent to some segment in B, we keep unchanged the
> information p(s),n(s),o(s) we started from.
>
> For the other squares, which are inside the circle but not on the
> circumferentce, we use an algorithm to interpolate the values
> p(s),n(s),o(s) (basically by averaging while keeping the circumference
> unchanged)
>
> Once we have computed the values p(s),n(s),o(s) for s in B, this is easy to
> produce the mesh.
>
> Laurent.
>
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