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Date: Mon, 27 Jun 1994 21:43:13 -0400 From: Tom Moertel Subject: Collision Detection - How?

Date: Mon, 4 Jul 1994 23:24:15 -0400 Subject: Typo fixed with 2K(K-1) expansion

Many people have requested copies of my collision detection code. I suspect that it's of general interest for the readers of this newsgroup, so I'm posting the code here along with a discussion of the techniques it uses. Please accept my apologies for the length of this posting.

The code was written in C++ on a Macintosh, but I've endeavored to keep the collision detection code close to ANSI C. Porting it should be a 30 minute affair. The testing-timing harness is C++- and Macintosh-specific, so it will take, say, an hour longer to port that, if you feel so inclined.


Here's how the code works, roughly speaking. The screen is divided into "sectors," defined by a regularly-spaced grid. All objects (e.g., sprites) are placed into the appropriate sectors as determined by the objects' upper-left corners. Then the objects in each sector are tested for collision with one another, taking advantage of the observation that overlapping objects will usually be classified into the same sector. This isn't always the case, however, and the code therefore makes well-behaved translations of the grid to ensure that all collisions will be detected and that no false collisions will be reported.


The first thing to do when you get the code is to look at the declaration of the "obj" structure. It represents an on-screen object. For convenience's sake, I've made all my objects 30x30. That way I can define the x and y data members to be the upper-left corner of an object's bounding rectangle, and when I need the lower-right, I calculate it by adding 30 to x and y. (That's the way I'd do it in a shoot-'em-up, too. Each class of objects would have a different size associated with it. E.g., for a bullet I'd add, say, 8 instead of 30 because they're smaller.)

I keep all the objects in a linked list, where the obj_link member is the link between objects. The sector_link is especially important. It is used to keep all the objects in a sector in a single linked list. That's a key to making this collision detection technique work quickly. Placing each object in its containing sector takes O(1) time, with a low constant, to boot.

With that in mind, here's an overview of the implementation:

  iterate four times, shifting the sector grid between iterations
      place objects into the appropriate sectors
      for each sector
          check for collisions among its objects

You may find it interesting that I've chosen to repeat the entire sectorization and per-sector collision checking process four times. That's how I get around the problems associated with overlapping objects that are placed into adjacent sectors. Instead of testing for collisions with objects in adjacent sectors, I just shift the entire sector grid and repeat the process. Before you accuse me of being insane for this "four-shifts" business, you should know that it's asymptotically 20 times faster than testing the adjacent sectors, and about 40 times faster for the most common "real world" cases. If you're interested in my analysis, it's near the end of my notes. Uninterested readers may feel free to skip it.

A side effect of the multiple iterations is that the same collision will sometimes be reported more than once. For example, if you have two objects directly on top of each other, they will both be placed in the same sector and detected as having collided, regardless of how the sector grid is shifted. The result: this particular collision will be reported four times. This isn't a big concern, and there are trivial ways to sidestep the issue, but I think I'd be remiss if I didn't point it out. I'd hate to have people screaming because particular bullets were packing four times the expected wallop, hurling their innocent spaceships into oblivion.


Before you begin thinking that this shift-and-repeat technique is terribly inefficient, consider the alternative, checking adjacent sectors. Let's say you've got a sector in the middle of the screen; call it S. Objects in S could collide with objects in adjacent sectors, so you'd have to include all eight of them in your collision testing of S. How does that affect running time?

Assume that objects are randomly distributed over the screen and that there are on average K objects in each sector. Recall that to test for collisions in each sector, we use a brute-force technique that requires n(n-1)/2 rectangle intersection operations (check it) for n objects. Now we can compare the four-shifts method with the test-adjacent-sectors method.

* Four-shifts method: each sector is checked by itself, at a cost of K(K-1)/2 rectangle tests, but the process is repeated 4 times. Consequently, the cost to entirely check a sector is 4 * K(K-1)/2 = 2K(K-1) = 2K^2 - 2K.

* Adjacent-sectors method: Each sector is checked only once, but its eight neighboring sectors are included in the check. Define L = (1+8)K be the average number of objects in these 9 sectors. So the cost per sector is L(L-1)/2 = (9K)1); cout « "Collision testing…" « endl; _RunExperiment( 0, false); _RunExperiment( 50, false); _RunExperiment(100, false); _RunExperiment(200, true ); draw this one } static void _RunExperiment(scalar numObjects, Boolean drawQ) { if (numObjects > kMaxObjects) return; too many cout « (int) numObjects « " objects: "; long endTime = Clock_us() + kTestLength; long iterations = 0; while (Clock_us() < endTime) { don't count initialization time { long t0 = Clock_us(); _RandomizeObjects(numObjects); endTime += Clock_us() - t0; } test/timing loop scalar i; for (i = 0; i < kCycleLength && Clock_us() < endTime; i++) _DetermineCollisions(), iterations++; } long totalTime = kTestLength + Clock_us() - endTime; if (drawQ) _ShowLastIteration(numObjects); draw results cout « (int) iterations « " in " « (int) totalTime « " us: "; float usec = totalTime; float iter = iterations; cout.precision(2); cout « usec/iter « " us/iter, " « 2) #define _IntersectQ(o1, o2) \ (_Abs(o1→x - o2→x) < kRectSize && \ _Abs(o1→y - o2→y) < kRectSize) #else inline scalar _Abs(scalar a) { return a < 0 ? -a : a; } inline scalar _IntersectQ(obj* o1, obj* o2) { return _Abs(o1→x - o2→x) < kRectSize && _Abs(o1→y - o2→y) < kRectSize; } #endif BRAIN_DEAD_INLINING static void _ClearCollisionArray() {

  memset(gCollisionArray, 0, sizeof(gCollisionArray));

} static void _CalcCollisionsInSector(obj* objList); static void _UpdateCollisionArray() {

  for (scalar x = 0; x < kNumXSectors; x++)
      for (scalar y = 0; y < kNumYSectors; y++)

} We've got the head of the linked list for a sector. Let's see if there are any objects in it that are involved in collisions. Use the plain, old O(n^2) technique to compute the collisions in this sector. If the grid size was appropriately chosen, n should be very small; in many cases it will be 0 or 1, obviating collision tests altogether. static void _CalcCollisionsInSector(obj* objList) { if (objList == NULL || objList→sector_link == NULL) return; for (obj* o0 = objList; o0→sector_link; o0 = o0→sector_link) for (obj* ox = o0→sector_link; ox; ox = ox→sector_link) if (_IntersectQ(o0, ox)) gCollisionArray[ o0 - gObjects ] = gCollisionArray[ ox - gObjects ] = 1; Note that at this point we know object o0

              // collided with object ox, so we could use that
              // information to, say, determine what kind of
              // explosion is appropriate.  Here, however, I
              // just toss the information away.

} ======= end Regards, Tom Moertel Interests: Software Engineering,

                                               Symbolic Mathematics,

MSA, CSG Technologies Division Algorithms, Itchy-Scratchy Theory.

9K)-1)/2 = (81K^2 - 9K)/2. Now, let's calculate the ratio of the two methods' expected number of rectangle tests:
          cost of adjacent-sectors   (81K^2 - 9K)/2
      R = ------------------------ = --------------
            cost of four-shifts         2K^2 - 2K
Note that the limit of R as K → Infinity is 20.25. Asymptotically, then, the four-shifts method is about 20 times faster than the adjacent-sectors method. Admittedly, it's unlikely you'll have an infinite number of objects on the screen. That fact begs the question, how much faster is the four-shifts method for the more common cases in which there are, on average, one, two, or three objects in a sector? Answer: For one object, it's *much* faster; for two, 38 x faster; for three, 30 x faster. The four-shifts method needs to perform *no* tests when there's only a single object in a sector—a very common case. The adjacent-sectors method, on the other hand, needs an average of 36 tests to handle the same situation. THE CODE Here it is. Enjoy. And, let me know how it works on your platform. If you port the testing-timing harness, please send me the timing results. The code is broken into sections. They are, in order:
  front matter        introductory comments
  declarations        defines constants and parameters
  test code           testing/timing harness (Mac specific)
  sector code         code that puts objects into sectors
  helpers             functions that are used by intersection code
  intersection code   uses sector and helper code to determine
                      object intersections and, hence, collisions
======= begin Sector-based collision detection routines & timing code. Tom Moertel 21-Jun-94 Results for a 25 MHz 68040 Macintosh (not exactly a screamer) and an 80 MHz PPC 601 Power Macintosh 8100 (this one screams): tests/s object count -68K- -PPC- 0 611 7640 50 340 4020 100 189 2060 200 81 788 where a "test" is defined to be a complete check of all objects, determining for each object whether it is involved in a collision (and if it is, with what other object). NOTES For this job I made all objects 30x30, but the code will work for arbitrarily-sized objects, with the restriction that objects are smaller than half of kSectorSize. This code is far from optimized. I didn't even bother to run it through a profiler. With a little work, it could probably be twice as fast. LEGAL STUFF Feel free to use this code in your own projects, but please give me credit. Copyright 1994 by Tom Moertel PORTING Most of the "real" code is portable C++, but the testing code uses some Mac- specific calls, namely Microseconds() and a few graphics and windowing calls. To port to the timing code to your platform, redifine Clock_us() to return the current state (count) of a fast internal clock in microseconds. The Macintosh drawing code will automaticaly compile out on non-Mac platforms, so if you want pretty pictures, you'll have to roll your own. #include <iostream.h> #include <string.h> #include <stdlib.h> #include <math.h> #if defined(macintosh) || defined(MWERKS) #include <Types.h> #include <Quickdraw.h> #include <Windows.h> #include <Events.h> #include <Timer.h> #endif define compilation parameters #if defined(MWERKS) || defined (SC) #define BRAIN_DEAD_INLINING define this to declare "hot" #endif functions as macros instead
                              // of C++ inline functions
define test parameters enum { kMaxObjects = 200, more than you're likely to need
  kRectSize       = 30,       // each object is 30 x 30 pixels
  kTBase          = 1000000L, // timing is in microseconds
  kTestLength     = 30*kTBase,// 30 seconds per experiment
  kCycleLength    = 50        // inner timing loop cycles 50 times
}; types #if defined(powerc) || defined (powerc) typedef int scalar; fast integer type #else typedef short scalar; fast integer type #endif sprite object struct obj { scalar x, y; coords obj* sector_link; link in sector list obj* obj_link; link in obj list … other members … } ; module-scope globals static obj gObjects[kMaxObjects]; static Boolean gCollisionArray[kMaxObjects]; forward declatations static void _DetermineCollisions(); static void _ShowLastIteration(scalar numObj); static void _RandomizeObjects(scalar numObj); static void _RunExperiment(scalar numObj, Boolean drawQ=false); ================================================================== test code ================================================================== returns a long representing a count of internal clock "ticks" #if defined(powerc) || defined (powerc) inline long Clock_us() { return TickCount() * (kTBase/60); } #else long Clock_us() { static UnsignedWide base; static Boolean initQ = true; if (initQ) Microseconds(&base), initQ = false; UnsignedWide x; Microseconds(&x); return (x.lo - base.lo); } #endif void main() { srand((unsigned int) Clock_us(
float)kTBase)*iter/usec « " iter/s" « endl; } ================================================================== sector code ================================================================== #define CEILING_DIV(x, y) ( ((x)+(y)-1) / (y) ) define constants Note that to work properly, kSectorSize must be greater than twice the length of the largest side of any object's bounding box. E.g., if your objects are 30x30, then the sector size should be > 60 – 64 would be an excellent choice. enum { kSectorSize = 64, length of a sector's side in pixels
  kLog2SectorSize =  6,   // log2(kSectorSize): for shifting
  kScreenWidth    = 640,
  kScreenHeight   = 480,
  kNumXSectors    = CEILING_DIV(kScreenWidth, kSectorSize) + 1,
  kNumYSectors    = CEILING_DIV(kScreenHeight, kSectorSize) + 1,
  kNumSectors     = kNumXSectors * kNumYSectors
} ; define a module-scope array of linked list heads, one for each sector static obj* gSectorArray[kNumXSectors][kNumYSectors]; call this routine to place all objects into the appropriate sectors (assumes all objects are kept in a linked list and GetMyFirstObject() returns the head of this list) extern obj* GetMyFirstObject(); static void UpdateSectors(register scalar xoff, register scalar yoff) { reset the sectors' linked lists
  obj** theArray = (obj**) gSectorArray; // for 1-D access
  for (scalar i = 0; i < kNumSectors; i++)
      *theArray++ = NULL;
  // put each object in its sector's linked list.
  for (obj* o = GetMyFirstObject(); o != NULL; o = o->obj_link)
      // get the list head for the sector in which o resides
      register obj** thisSectorListHead =
          &gSectorArray [ (o->x + xoff) >> kLog2SectorSize ]
                        [ (o->y + yoff) >> kLog2SectorSize ];
      // add o to this sector's linked list
      o->sector_link = *thisSectorListHead;
      *thisSectorListHead = o;
} ================================================================== helpers ================================================================== Draw an object (rectangle). If the object is involved in a collision, it is drawn as a rectanglular outline; otherwise it's drawn as a solid gray rectangle. [Macintosh specific] static void _DrawObject(obj* o, Boolean collidedQ) { #if defined(macintosh) || defined(MWERKS) static Pattern myBlack = { 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff }; static Pattern myGray = { 0xaa, 0x55, 0xaa, 0x55, 0xaa, 0x55, 0xaa, 0x55 }; Rect r; SetRect(&r, o→x, o→y, o→x + kRectSize, o→y + kRectSize); PenPat(collidedQ ? &myBlack : &myGray); if (collidedQ) FrameRect(&r); else PaintRect(&r); #endif macintosh } conciliate skeptics by showing them that the code did, indeed, work properly [Macintosh specific] static void _ShowLastIteration(scalar numObjects) { #if defined(macintosh) || defined(MWERKS) Rect rBounds = { 0, 0, kScreenHeight, kScreenWidth }; OffsetRect(&rBounds, 0, GetMBarHeight()); WindowPtr wind = NewWindow(nil, &rBounds, "\p", true, plainDBox, WindowPtr(-1), false, 0); GrafPtr savePort; GetPort(&savePort); SetPort(wind); for (scalar i = 0; i < numObjects; i++) _DrawObject(&gObjects[i], gCollisionArray[i]); while (!Button()) ; SetPort(savePort); DisposeWindow(wind); #endif macintosh } static scalar _RandScalar(scalar max) {
  return (((unsigned long) max) *
          ((unsigned short) rand())) / (RAND_MAX+1);
} static void _RandomizeObjects(scalar numObjects) {
  obj* o = gObjects;
  for (scalar i = 0; i < numObjects; i++, o++)
      o->x        = _RandScalar(kScreenWidth-1);
      o->y        = _RandScalar(kScreenHeight-1);
      o->obj_link = o + 1;
  (--o)->obj_link = NULL;
} ================================================================== intersection code ================================================================== obj* GetMyFirstObject() { return &gObjects[0]; } local helpers static void _ClearCollisionArray(); static void _UpdateCollisionArray(); determine all collisions static void _DetermineCollisions() { _ClearCollisionArray(); erase the slate; no collisions yet
  scalar shift = kSectorSize / 2;
  // We need to try four differnt "shifts" of the
  // sector grid to detect all collisions.  Proof of
  // why this is so is left as an excercise for the
  // reader.  (Hint: consider an analogous 1-D case.)
  UpdateSectors(    0,     0),    _UpdateCollisionArray();
  UpdateSectors(    0, shift),    _UpdateCollisionArray();
  UpdateSectors(shift,     0),    _UpdateCollisionArray();
  UpdateSectors(shift, shift),    _UpdateCollisionArray();
} "hot" functions that are used in inner loops #ifdef BRAIN_DEAD_INLINING #define _Abs(a) ((a) < 0 ? -(a) : (a
/data/webs/external/dokuwiki/data/pages/archive/games/collisio.txt · Last modified: 2002/05/06 15:02 by

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