checking whether 2 2D polygons overlap

checking whether 2 2D polygons overlap

Post by Hugo Lyppe » Wed, 05 Dec 1990 20:17:49



I've partly implemented the Newell-Newell-Sancha hidden surface algorithm on the Amiga for a real-time 3D animation package I've written. This algorithm is also known as the Painter's algorithm, for it establishes a priority list and draws polygons from the back to the front.

The program allows scenes composed of planar polygons which have no holes. I
don't however get perfect results because there's one step in the algorithm I
 don't know how to implement: to check if the projections of two polygons on
the screen overlap or not. I haven't been able to find an article on this
subject, not even in articles from the authors of the painter's algorithm
themselves!

So please could anyone help me with some hints or source code to determine QUICKLY whether two 2D polygons overlap or not? Then I'll be able to implement the algorithm completely.

Please E-mail your responses to


Any help is greatly appreciated!

 
 
 

checking whether 2 2D polygons overlap

Post by Richard No » Thu, 06 Dec 1990 09:43:10


[implementing newell-newell-sancha on an Amiga]

Quote:>the screen overlap or not. I haven't been able to find an article on this
>subject, not even in articles from the authors of the painter's algorithm
>themselves!

I did this a while ago, using Foley-VanDam.  The 2D-overlap algo is either
in the same book, or another common reference, because I didn't have
trouble finding it.  The only problem is, this algorithm DOESN'T WORK
CONSISTENTLY.  Even if you limit yourself to scenes composed of the simplest
non-intersecting triangles, newell-newell-sancha algo will occasionally
be unable to depth-sort them correctly.

Is there an algorithm that always works?  If there is, it's a closely
gaurded secret of the high priests of computer graphics.  I've tried for
years to find an algorithm that will work in all cases, spending months
coding, testing, and ending up in defeat.

I went to Computer Literacy (for those unknowing, it's a VERY complete
bookstore) and looked in EVERY computer graphics text they had.
Many didn't even mention hidden-line algos, some mentioned them but didn't
elaborate, and several only had elaborate rendering techniques useless
for real-time.

Only two books (everybody knows which) had a depth-sort algorithm in
detail, and they both described the (oh no!) Newell-Newell-Sancha algo.

If anyone could mention/describe/point to a SINGLE ALGORITHM ANYWHERE
that depth-sorts correctly, I would be indebted forever.  Would one of
the high priests reading this group care to reveal this secret to the
masses?

---------------------------------
Ben Discoe, caltech escapee and frustrated visionary at large.

 
 
 

checking whether 2 2D polygons overlap

Post by Jeffrey P. Bak » Thu, 06 Dec 1990 14:57:31




> [implementing newell-newell-sancha on an Amiga]
> >the screen overlap or not. I haven't been able to find an article on this
> >subject, not even in articles from the authors of the painter's algorithm
> >themselves!

[ Etc ... ]
> If anyone could mention/describe/point to a SINGLE ALGORITHM ANYWHERE
> that depth-sorts correctly, I would be indebted forever.  Would one of
> the high priests reading this group care to reveal this secret to the
> masses?

Well, I'm not a graphics Guru of any version but I really don't think there
is any way to have a depth sort algorithm work in all cases.  The depth
sort always fails whenever you have intersecting polygons or polygons that
overlay sections of each other.  I think the only way to work around this
problem short of ray-tracing or some other high cpu cost procedure is the
use of the Z-buffer algorithm (The painters is a simplified version of this
algorithm) when you generally depth sort each pixel individually instead of
each polygon.  This is all that I have ever seen to eliminate this problem
short of highly-intensive algorithms.  Only problem of course with the
Z buffer algorithm is that you need a fairly large amount of memory
to store the necessary information.  Although if you have the memory, there
would be very few ways of doing this any faster with a different algorithm.

Anyone else out there want to take a shot at a better solution.

--
Jeffrey P. Bakke                      |   There are a finite number of

  UUCP    : ...!uunet!plains!bakke    |     The overflow began

"I am not a number, I am a free man!" - The Prisoner

 
 
 

checking whether 2 2D polygons overlap

Post by Tom Du » Thu, 06 Dec 1990 22:57:01


Checking two polygons for overlap is fairly easy:
either some edge of one polygon crosses an edge of the other,
or one of the polygons is completely contained within the other.

You can check for containment by picking a single vertex of each
polygon and checking it for containment in the other.  (You
know how to do this, right?  If not, check the FAQs.)  If these
tests fail, you can just test each pair of edges for intersection.
The obvious method of doing this takes time proportional to n^2.
Since two polygons can intersect in O(n^2) points, you probably
can't do better than this, worst case.

Of course, as has been already pointed out, a bounding box cull
will usually save much work.

The previous posting that suggested using a general polygon clipper
(like Weiler-Atherton) to test for overlap was overkill.  It's
Very Hard to get Weiler-Atherton to work properly; it's dead easy
to check polygons for overlap.

 
 
 

checking whether 2 2D polygons overlap

Post by John Schul » Thu, 06 Dec 1990 04:04:20



Quote:>I've partly implemented the Newell-Newell-Sancha hidden surface algorithm on the Amiga for a real-time 3D animation package I've written. This algorithm is also known as the Painter's algorithm, for it establishes a priority list and draws polygons from the back to the front.
>The program allows scenes composed of planar polygons which have no holes. I
>don't however get perfect results because there's one step in the algorithm I
> don't know how to implement: to check if the projections of two polygons on
>the screen overlap or not. I haven't been able to find an article on this
>subject, not even in articles from the authors of the painter's algorithm
>themselves!
>So please could anyone help me with some hints or source code to determine QUICKLY whether two 2D polygons overlap or not? Then I'll be able to implement the algorithm completely.

  Two steps: Check to see if their extents overlap (minX,maxX,minY,maxY of
the polygon). This step requires only boolean comparisons. You can compute
the extents in your 2D clipping code. If you're in for speed, this step
may be all you need.
  The next step, if the extents overlap, is to find out if the polygons
actually overlap. One way is to use a clipping algorithm to clip one
polygon against another (Instead of clipping against a rectangle, clip
against a polygon. Most screen clippers are just special case clippers
for rectangles). You might also be able to analyze the relationship
between the area of the extent of *both* polygons and the areas of each
polygon within that extent (overlapping polygons will have less area than
non-overlapping polygons).

  John

 
 
 

checking whether 2 2D polygons overlap

Post by Alvin T. Campbell I » Fri, 07 Dec 1990 02:06:59


In article 15436 of comp.graphics, (Jeffrey P. Bakke) writes:


>> If anyone could mention/describe/point to a SINGLE ALGORITHM ANYWHERE
>> that depth-sorts correctly, I would be indebted forever.

>                                    ... I really don't think there
>is any way to have a depth sort algorithm work in all cases.  The depth
>sort always fails whenever you have intersecting polygons or polygons that
>overlay sections of each other.  I think the only way to work around this
>problem ... is the use of the Z-buffer algorithm

>Anyone else out there want to take a shot at a better solution.

I would suggest the use of a Binary Space Partioning (BSP) Tree algorithm.
The basic technique leads to a depth sort without the limitations of the
Newell-Newell-Sancha method, mentioned above.  

First, the input polygons are split across each other's planes, in order to
prevent the intersection and cycled-ordering problems mentioned above.  This
is done in an efficient manner which does not require each polygon to be
tested against all others.

Next, a binary data structure is created which aids in the polygon
sorting process.  A modified inorder tree traversal results in a correct
depth sort of the polygons from any viewpoint.  Note that this data
structure is viewpoint-independent, and is well-suited for hidden
surface removal in animations where the only the viewpoint changes.

The basic work was done by Bruce Naylor in his dissertation research,
and is documented in the somewhat theoretical paper:
Fuchs, Kedem, and Naylor, "On Visible Surface Generation by A Priori
Tree Structures", Proceedings of SIGGRAPH 1980, July 1980, pp. 124-133.

A later paper with a more practical orientation is:
Fuchs, Abram, and Grant, "Near Real-Time Shaded Display of Rigid
Objects", Proceedings of SIGGRAPH 1983, JUly 1983, pp. 65-72.

I hope this information is helpful.

--
                                A. T. Campbell, III
                                CS Department, University of Texas

 
 
 

checking whether 2 2D polygons overlap

Post by Samuel P. Uselt » Fri, 07 Dec 1990 02:52:12




>[implementing newell-newell-sancha on an Amiga]
>>the screen overlap or not. I haven't been able to find an article on this
>>subject, not even in articles from the authors of the painter's algorithm
>>themselves!

As pointed out by others (and my email to Lyppens) this is really the
same as clipping - found in lots of books.

Quote:>I did this a while ago, using Foley-VanDam.  The 2D-overlap algo is either
>in the same book, or another common reference, because I didn't have
>trouble finding it.  The only problem is, this algorithm DOESN'T WORK

                                           ^^^^^^^^^^^^^^
Newell-Newell-Sancha or the overlap (clipping) ??
(I assume N-N-S Painter's visible surface alg.)

Quote:>CONSISTENTLY.  Even if you limit yourself to scenes composed of the simplest
>non-intersecting triangles, newell-newell-sancha algo will occasionally
>be unable to depth-sort them correctly.

There is a standard picture in most texts showing a set of three convex
polygons that overlap in x y and z without intersecting.  They can not be
"properly" depth sorted BY ANY METHOD! (So don't blame N-N-S or the text
authors.)  The standard solution in this case is to split one of the polygons
along the plane of another.  This results in four convex polygons that ARE
readily depth "sortable".

The main benefit of this algorithm is not its speed for a single frame but
rather the fact that in most SEQUENCES the order of the polygons changes
very little from one frame to the next, so frames 2 through N can be done
faster.  This speed is useful in real (or near real) time work, but requires
fairly clever implementation to take full advantage.

Quote:

>Is there an algorithm that always works?  If there is, it's a closely

          ^^^^^^^^^^^^  "Hidden Surface Removal" I assume

Quote:>gaurded secret of the high priests of computer graphics.  I've tried for
>years to find an algorithm that will work in all cases, spending months
>coding, testing, and ending up in defeat.

There is no ONE visible surface rendering algorithm best for all purposes.
That is why there are so many choices.  And why people ("high priests"? :-) )
continue to do research and publish new algorithms and improvements to the
old ones.  I'm only a low priest I guess (  :-)  ) but in every graphics
class I ever taught (in 10 years) we spent at least a couple of weeks
discussing the hidden/visible surface problem, why there were choices to be
made, and what the question were to help determine which answer is appropriate.
Hardly a "closely guarded secret."

Quote:

>I went to Computer Literacy (for those unknowing, it's a VERY complete
>bookstore) and looked in EVERY computer graphics text they had.
>Many didn't even mention hidden-line algos, some mentioned them but didn't
>elaborate, and several only had elaborate rendering techniques useless
>for real-time.

It is a HARD problem.  Saying "useless for real-time" and discounting
the entire content is throwing the baby out with the bathwater.  It also
suggests that your hardware is not (yet) up to the task you are attempting.
Quote:

>Only two books (everybody knows which) had a depth-sort algorithm in
>detail, and they both described the (oh no!) Newell-Newell-Sancha algo.

>If anyone could mention/describe/point to a SINGLE ALGORITHM ANYWHERE
>that depth-sorts correctly, I would be indebted forever.  Would one of

^^^^^^^^^^^^^^^^^^^^^^^^^^^ as pointed out above,
this is mathematically impossible in general.

Quote:>the high priests reading this group care to reveal this secret to the
>masses?

Another possibility you might consider  (before buying a Personal Iris
and making the problem moot) is the BSP-tree based algorithm.  It requires
building a tree data structure rather than a sorted list.  Traversing the
tree at rendering time is almost as fast.  The tree does not need changed
as a viewer moves around, but will need revised when objects move relative to
each other.  [Foley, VanDam, Feiner & Hughes starting p. 675]

Quote:>---------------------------------
>Ben Discoe, caltech escapee and frustrated visionary at large.


employed by CSC         working for NASA        speaking for myself
 
 
 

checking whether 2 2D polygons overlap

Post by Tom Du » Fri, 07 Dec 1990 23:48:07



Quote:>Even if you limit yourself to scenes composed of the simplest
>non-intersecting triangles, newell-newell-sancha algo will occasionally
>be unable to depth-sort them correctly.
>Is there an algorithm that always works?  If there is, it's a closely
>gaurded secret of the high priests of computer graphics.  I've tried for
>years to find an algorithm that will work in all cases, spending months
>coding, testing, and ending up in defeat.

I find this hard to believe.  Foley, van Dam, Feiner & Hughes describe
the Newell, Newell & Sancha algorithm on pp 673-675 of Computer Graphics,
Principles & Practice.  The algorithm is fairly simple.
I've coded it up at least once, long ago when I gave it as a programming
assignment.  It took about a day, and
never caused any trouble.  I'm utterly astounded that anyone could spend
large amounts of time and still fail to get it working.

It's fairly obvious that the algorithm cannot fail to depth-sort a scene,
since whenever the depth order of two polygons cannot be determined, it
replaces one of them by splitting it in two pieces, one on each side of the
other (original) polygon.  Since the algorithm refuses to allow polygons
whose depth order cannot be determined to remain in the scene, the only
possible failure mode would be non-termination, not incorrect depth-sorting.
Non-termination is obviously out of the question, since each polygon can
be split at most by each other (original) polygon in the scene.

Maybe you could describe in more detail a simple scene on which your program
fails?

 
 
 

checking whether 2 2D polygons overlap

Post by Daniel Mark Gess » Sat, 08 Dec 1990 02:02:19



>Actually, the Z-buffer is a very brute force technique, one might even say "stupid".
>It get's the job done, but it wastes a lot of time dealing with every pixel from
>every polygon in the scene, whether or not it will be ultimately visible.  It is
>also is not compatible with any kind of real-time anti-aliasing technique.  The
>solution, of course, involves much more complex algorithms which perform various
>types of culling operations in order to limit the number of pixels actually rendered.

                                                   Todd Heckel, net neophyte.

Actually, anti-aliasing a zbuffer is easy. But the technique I would use is as
brute force and stupid as a z-buffer itself. Render the image into a buffer
with twice the resolution in each direction (or more) and have video hardware
that averages from 4 pixels (or more). This is probably the most
general anti-aliasing technique, and I'd like to see it in things like display
postscript and stuff. But it does use 4 (or more) times as much memory and
processing power.

Dan
--
Daniel Mark Gessel                          Independent Software Consultant

I do not represent Swarthmore College (thank God).

 
 
 

checking whether 2 2D polygons overlap

Post by Todd Heck » Fri, 07 Dec 1990 23:15:12


Actually, the Z-buffer is a very brute force technique, one might even say "stupid".
It get's the job done, but it wastes a lot of time dealing with every pixel from
every polygon in the scene, whether or not it will be ultimately visible.  It is
also is not compatible with any kind of real-time anti-aliasing technique.  The
solution, of course, involves much more complex algorithms which perform various
types of culling operations in order to limit the number of pixels actually rendered.

                                                Todd Heckel, net neophyte.

 
 
 

checking whether 2 2D polygons overlap

Post by John Schul » Sat, 08 Dec 1990 10:51:48



[stuff deleted]

Quote:>Is there an algorithm that always works?  If there is, it's a closely
>gaurded secret of the high priests of computer graphics.  I've tried for
>years to find an algorithm that will work in all cases, spending months
>coding, testing, and ending up in defeat.

  The BSP (binary space partition algorithm) tree will work for all cases.
The only disadvantage is the need to split polygons with intersecting planes.
These splits are done *before* runtime (for real time work), so this algorithm
is very good when you can crank out polygons at high speed.
  You can also do splits for the Newell et all algorithm *during* runtime,
and thus is will work for all cases (but slower).

Quote:>If anyone could mention/describe/point to a SINGLE ALGORITHM ANYWHERE
>that depth-sorts correctly, I would be indebted forever.  Would one of
>the high priests reading this group care to reveal this secret to the
>masses?

  See "Computer Graphics, Principles and Practice", by Foley et all.
It presents a lucid explanation of BSPs and the depth sort method with
splits.

  John

 
 
 

checking whether 2 2D polygons overlap

Post by John Schul » Sat, 08 Dec 1990 11:06:57



[stuff deleted]

Quote:>The previous posting that suggested using a general polygon clipper
>(like Weiler-Atherton) to test for overlap was overkill.  It's
>Very Hard to get Weiler-Atherton to work properly; it's dead easy
>to check polygons for overlap.

  The Sutherland-Hodgman algorithm will clip any polygon (convex or
concave) to any convex polygon. It's really simple to code, and can
also be used as the primary 2D or 3D screen clipper. But, doing a
pointwise compare (using cross products or winding points algorithms),
of points in a polygon will probably be faster. But, if you're out for
speed, skip this step, or if you have a fast polygon machine, use a BSP
(or if objects are moving, use BSPs for each object, then sort the objects).

  John

 
 
 

checking whether 2 2D polygons overlap

Post by Ian Kemmi » Sun, 09 Dec 1990 11:01:34


<Discussion of overlapping polygons in the depth sort. >

There are various ways to do this.  The A-buffer fudges it, on the
basis that in most cases the error introdiced by fudging isv.
small.  (And I've known plenty of commercial work done this way
that seems to bear this out.)  The Warnock subdivision algorithm
would certainly handle it OK, until you got to the sub-pixel level,
when you could again fudge it.  I beleive the Weiler-Atherton algorithm
handles it if done properly.  Catmull's analytic algorithm (SIGGRAPH 84)
certainly does it properly.

I am surprised at the claim that no textbooks could be found that
mentioned these algorithms.

Also, note that `doing it properly' is not necessarily expensive.  Most
of these algorithms were devised when the VAX was new.  These days,
a flop is far cheaper than a page fault -- I am in the middle of
moving from a super-sampled jittered z-buffer to an analytic
visibility algorithm with direct-convolution filtering for just
that reason (plus a natural fear of patent lawyers!).

--
Ian D. Kemmish                    Tel. +44 767 601 361

Biggleswade                       ukc!dircon!uad1077

 
 
 

checking whether 2 2D polygons overlap

Post by Jeff Bak » Sat, 08 Dec 1990 18:40:39



> Actually, the Z-buffer is a very brute force technique, one might even say "stupid".
> It get's the job done, but it wastes a lot of time dealing with every pixel from
> every polygon in the scene, whether or not it will be ultimately visible.  It is
> also is not compatible with any kind of real-time anti-aliasing technique.  The
> solution, of course, involves much more complex algorithms which perform various
> types of culling operations in order to limit the number of pixels actually rendered.

Correct me if I'm wrong but from what I understand the Z-Buffer method,
is the method most implemented in High performance hardware graphics
implementation.  On the hardware level it would seem to be very efficient
and easy to implement.  I agree that on a software level, it would be
stupid to fully implement a z-buffer method of hl removal but I made the
assumption that if you were to do a software method, you would make
adjustments and changes to the generalized algorithm to decrease unnecesary
calculation and overlap.

--
Jeffrey P. Bakke                      |   There are a finite number of

  UUCP    : ...!uunet!plains!bakke    |     The overflow began

"I am not a number, I am a free man!" - The Prisoner

 
 
 

checking whether 2 2D polygons overlap

Post by Todd Heck » Mon, 10 Dec 1990 07:23:18



> Actually, anti-aliasing a zbuffer is easy. But the technique I would use is as
> brute force and stupid as a z-buffer itself. Render the image into a buffer
> with twice the resolution in each direction (or more) and have video hardware
> that averages from 4 pixels (or more). This is probably the most
> general anti-aliasing technique, and I'd like to see it in things like display
> postscript and stuff. But it does use 4 (or more) times as much memory and
> processing power.

> Dan
> --
> Daniel Mark Gessel                          Independent Software Consultant

> I do not represent Swarthmore College (thank God).

        Note that I said "real-time".  I have designed systems that do exactly
        as you described, at either four or six* times the display resolution
        (two by two or four by four).  While the images look great, they take
        four or (gasp) six* times as long to generate as an un-anti-aliased
        image.  Newer techniques based on sorting and bit-masks, like the
        A-buffer, are currently being adapted for real-time image generation
        use.
--
        Todd Heckel
        Megatek Corporation     uunet!megatek!toddh    
        9645 Scranton Road              or

 
 
 

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