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Date: Mon, 27 Jun 1994 21:43:13 -0400
From: Tom Moertel <thor@telerama.lm.com>
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.
OVERVIEW
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.
NOTES
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.
ANALYSIS: FOUR-SHIFTS vs. ADJACENT-SECTORS
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)((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
// moertel@acm.org
//
// 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());
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, "
<< ((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))
#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++)
_CalcCollisionsInSector(gSectorArray[x][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,
thor@telerama.lm.com Itchy-Scratchy Theory.