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Copy path[32-5] [Blog 1999] Simple Intersection Tests for Games.html
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[32-5] [Blog 1999] Simple Intersection Tests for Games.html
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<!DOCTYPE html> <html lang=en class=hb-loaded style><!--
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url: https://www.gamasutra.com/view/feature/131790/simple_intersection_tests_for_games.php?page=5
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<title>Gamasutra - Simple Intersection Tests For Games</title>
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<meta name=keywords content="game development, game developer, game programming, game programmer, videogame, artificial intelligence, 3D animation, game design, 3D modeling, game business, game jobs, game directory, game development software, game technology, 3D technology, game producer, game audio, game animation, virtual reality, digital entertainment, PC game, Xbox game, game news, new game, arcade development, Nintendo development, Playstation development, Playstation 2, PS2, Dreamcast development, Game Developer magazine, Computer Game Developers Conference, Game Developers Conference, Independent Game Developers Conference, CMP Game Media Group, game industry research, online game development, digital assets, free textures, free 3D models, free shaders, Gamasutra Exchange, 3D Studio Max textures"><meta name=description content="A key component to virtually any graphical game, 2D or 3D, is a collision detection system for game objects. In this article, Looking Glass's Miguel G"><link rel=image_src href=https://twimgs.com/gamasutra//images/questionmark.jpg>
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" 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<strong>Simple Intersection Tests For Games</strong>
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<span style=color:#660000> Page 5 </span>
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" 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<h1 class=item_title>Simple Intersection Tests For Games</h1>
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<span class=newsAuth>by <a href=https://www.gamasutra.com/view/authors/751407/miguel_gomez.php>miguel gomez</a></span><span class=pageNav> [<span class=category_links><a class=category_links href=https://www.gamasutra.com/features//></a></span>]</span><br>
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<title>Error 404 (Not Found)!!1</title>
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<p><b>404.</b> <ins>That’s an error.</ins>
<p>The requested URL <code>/u/0/se/0/_/+1/fastbutton</code> was not found on this server. <ins>That’s all we know.</ins>
</p>'></iframe></div><g:plusone size=medium count=false data-gapiscan=true data-onload=true data-gapistub=true></g:plusone>
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<a href=http://feeds.feedburner.com/GamasutraFeatureArticles><img src=data:image/gif;base64,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 width=15 height=15 border=0 alt=RSS></a>
</div>
<hr>
<strong>October 18, 1999</strong>
<a href=https://www.gamasutra.com/view/feature/131790/simple_intersection_tests_for_games.php><img src='data:image/svg+xml,<svg xmlns="http://www.w3.org/2000/svg" width="91" height="16"><rect fill-opacity="0"/></svg>' width=91 height=16 border=0 alt="Article Start" style="background-blend-mode:normal!important;background-clip:content-box!important;background-position:50% 50%!important;background-color:rgba(0,0,0,0)!important;background-image:var(--sf-img-38)!important;background-size:100% 100%!important;background-origin:content-box!important;background-repeat:no-repeat!important"></a>
<a href="https://www.gamasutra.com/view/feature/131790/simple_intersection_tests_for_games.php?page=4">
<img src='data:image/svg+xml,<svg xmlns="http://www.w3.org/2000/svg" width="68" height="16"><rect fill-opacity="0"/></svg>' width=68 height=16 border=0 alt=Previous style="background-blend-mode:normal!important;background-clip:content-box!important;background-position:50% 50%!important;background-color:rgba(0,0,0,0)!important;background-image:var(--sf-img-39)!important;background-size:100% 100%!important;background-origin:content-box!important;background-repeat:no-repeat!important">
</a>
<strong>Page 5 of 7</strong>
<a href="https://www.gamasutra.com/view/feature/131790/simple_intersection_tests_for_games.php?page=6">
<img src='data:image/svg+xml,<svg xmlns="http://www.w3.org/2000/svg" width="44" height="16"><rect fill-opacity="0"/></svg>' width=44 height=16 border=0 alt=Next style="background-blend-mode:normal!important;background-clip:content-box!important;background-position:50% 50%!important;background-color:rgba(0,0,0,0)!important;background-image:var(--sf-img-40)!important;background-size:100% 100%!important;background-origin:content-box!important;background-repeat:no-repeat!important">
</a>
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<div class="item_body mobile_image_transform">
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<p align=left>
<strong>An
Oriented Bounding Box (OBB) Intersection Test</strong>
</p>
<p align=left>
A
drawback of using an axis-aligned bounding box is that it can’t fit
rotating geometry very tightly.
</p>
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<div class="module_right right bottom30">
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<p align=left>
On
the other hand, an oriented bounding box can be rotated with the objects,
fitting the geometry with less volume than an AABB. This requires that
the orientation of the box must also be specified. Figure 8 shows a
2D example, where <strong>A</strong><sup>1</sup>, <strong>A</strong><sup>2</sup>, <strong>B</strong><sup>1</sup>
and <strong>B</strong><sup>2</sup> are the local axes of boxes A and B.
</p>
<p align=left>
</p>
<p align=left>
</p>
<p align=left>
</p>
<p align=left>
<img src="data:image/gif;base64,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" alt width=430 height=198>
</p>
<p align=left>
For
OBBs, the separating axis test must be generalized to three dimensions.
A box's scalar projection onto a unit vector <strong>L</strong> creates an interval
along the axis defined by <strong>L</strong>.
</p>
<p align=left>
<img src=data:image/gif;base64,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 alt width=398 height=244>
</p>
<p align=left>
The
radius of the projection of box A onto <strong>L</strong> is
</p>
<p align=left>
<img src=data:image/gif;base64,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 alt width=258 height=46>
</p>
<p align=left>
The
same is true for B, and <strong>L</strong> forms a separating axis if
</p>
<p align=left>
<img src="data:image/gif;base64,R0lGODlhbgAgAID/AMDAwAAAACH5BAEAAAAALAAAAABuACAAAAK9hI+py+0Po5y02ouz3rz7D4biSJbmiaaqGpDtir6iDJdtgOf4o+8JbQDWNrfgS7jwKZDI4aUI8DV/zSpoGqIpI1vEFFsBf7TfpbXBvImDjF0XQ0ab27wkFN0TujvxyZstN0dRtebnVWhEp7hkcTZ2OKh14MgYthiFePmH5zWZFMhXmXHUA5GX4/njoLe5SpUpmsVxNxqlAWt4C1hryztD1KuLqzoyfIiqC1zsxNzs/AwdLT1NXW19jZ2tHVIAADs=" alt width=110 height=32>
</p>
<p align=left>
Note
that <strong>L</strong> does not have to be a unit vector for this test to work.
The boxes A and B are disjoint if none of the 6 principal axes and their
9 cross products form a separating axis. These tests are greatly simplified
if <strong>T</strong> and B’s basis vectors (<strong>B</strong><sup>1</sup>, <strong>B</strong><sup>2</sup>,
<strong>B</strong><sup>3</sup>) are transformed into A’s coordinate frame.
</p>
<p align=left>
An
OBB class and an implementation of the OBB overlap test is given in
Listing 6 below.
</p>
<table border=0 width=100%>
<tbody>
<tr align=left>
<td width=450>
<p align=left>
<strong><em>Listing
6. An OBB class.</em></strong>
</p>
<p align=left>
#include
"coordinate_frame.h"
</p>
<p align=left>
class
OBB : public COORD_FRAME<br>
{
</p>
<p align=left>
public:
</p>
<p align=left>
</p>
<div align=left>
<blockquote>
VeCTOR
E; //extents<br>
OBB(
const VECTOR& e ): E(e)
</blockquote>
</div>
<blockquote>
<p align=left>
{}
</p>
</blockquote>
<p align=left>
};
</p>
<p align=left>
//check
if two oriented bounding boxes overlap<br>
const
bool OBBOverlap<br>
(
</p>
<p align=left>
</p>
<div align=left>
<blockquote>
//A<br>
VECTOR& a, //extents<br>
VECTOR& Pa, //position<br>
VECTOR* A, //orthonormal
basis
</blockquote>
</div>
<blockquote>
<p align=left>
//B<br>
VECTOR& b, //extents<br>
VECTOR& Pb, //position<br>
VECTOR* B //orthonormal
basis
</p>
</blockquote>
<p align=left>
)
</p>
<p align=left>
{
</p>
<p align=left>
</p>
<div align=left>
<blockquote>
//translation,
in parent frame<br>
VECTOR
v = Pb - Pa;
</blockquote>
</div>
<blockquote>
<p align=left>
//translation,
in A's frame<br>
VECTOR
T( v.dot(A[0]), v.dot(A[1]), v.dot(A[2]) );
</p>
<p align=left>
//B's
basis with respect to A's local frame<br>
SCALAR
R[3][3];<br>
float
ra, rb, t;<br>
long
i, k;
</p>
<p align=left>
//calculate
rotation matrix<br>
for(
i=0 ; i<3 ; i++ )
</p>
<p align=left>
</p>
<div align=left>
<blockquote>
for(
k=0 ; k<3 ; k++ )
</blockquote>
</div>
<blockquote>
<p align=left>
</p>
<div align=left>
<blockquote>
R[i][k]
= A[i].dot(B[k]);
</blockquote>
</div>
<blockquote>
<p align=left>
/*ALGORITHM:
Use the separating axis test for all 15
potential <br>
separating axes. If a separating
axis could not be found, the two <br>
boxes overlap. */
</p>
<p align=left>
//A's
basis vectors<br>
for(
i=0 ; i<3 ; i++ )<br>
{
</p>
<p align=left>
</p>
<div align=left>
<blockquote>
<p>
ra
= a[i];
</p>
<p>
rb
= <br>
b[0]*fabs(R[i][0]) + b[1]*fabs(R[i][1]) + b[2]*fabs(R[i][2]);
</p>
</blockquote>
</div>
<blockquote>
<p align=left>
t
= fabs( T[i] );
</p>
<p align=left>
if(
t > ra + rb ) <br>
return false;
</p>
</blockquote>
<p align=left>
}
</p>
<p align=left>
//B's
basis vectors<br>
for(
k=0 ; k<3 ; k++ )<br>
{
</p>
<p align=left>
</p>
<blockquote>
ra
= <br>
a[0]*fabs(R[0][k]) + a[1]*fabs(R[1][k]) + a[2]*fabs(R[2][k]);
</blockquote>
<blockquote>
<p align=left>
rb
= b[k];
</p>
<p align=left>
t
= <br>
fabs( T[0]*R[0][k] + T[1]*R[1][k] + <br>
T[2]*R[2][k] );
</p>
<p align=left>
if(
t > ra + rb )<br>
return
false;
</p>
</blockquote>
<p align=left>
}
</p>
<p align=left>
//9
cross products<br>
<br>
//L
= A0 x B0<br>
ra
= <br>
a[1]*fabs(R[2][0]) + a[2]*fabs(R[1][0]);
</p>
<p align=left>
rb
= <br>
b[1]*fabs(R[0][2]) + b[2]*fabs(R[0][1]);
</p>
<p align=left>
t
= <br>
fabs( T[2]*R[1][0] - <br>
T[1]*R[2][0] );
</p>
<p align=left>
if(
t > ra + rb )<br>
return
false;
</p>
<p align=left>
//L
= A0 x B1<br>
ra
= <br>
a[1]*fabs(R[2][1]) + a[2]*fabs(R[1][1]);
</p>
<p align=left>
rb
= <br>
b[0]*fabs(R[0][2]) + b[2]*fabs(R[0][0]);
</p>
<p align=left>
t
= <br>
fabs( T[2]*R[1][1] - <br>
T[1]*R[2][1] );
</p>
<p align=left>
if(
t > ra + rb )<br>
return
false;
</p>
<p align=left>
//L
= A0 x B2<br>
ra = <br>
a[1]*fabs(R[2][2]) + a[2]*fabs(R[1][2]);
</p>
<p align=left>
rb
= <br>
b[0]*fabs(R[0][1]) + b[1]*fabs(R[0][0]);
</p>
<p align=left>
t
= <br>
fabs( T[2]*R[1][2] - <br>
T[1]*R[2][2] );
</p>
<p align=left>
if(
t > ra + rb )<br>
return
false;
</p>
<p align=left>
//L
= A1 x B0<br>
ra
= <br>
a[0]*fabs(R[2][0]) + a[2]*fabs(R[0][0]);
</p>
<p align=left>
rb
= <br>
b[1]*fabs(R[1][2]) + b[2]*fabs(R[1][1]);
</p>
<p align=left>
t
= <br>
fabs( T[0]*R[2][0] - <br>
T[2]*R[0][0] );
</p>
<p align=left>
if(
t > ra + rb )<br>
return
false;
</p>
<p align=left>
//L
= A1 x B1<br>
ra
= <br>
a[0]*fabs(R[2][1]) + a[2]*fabs(R[0][1]);
</p>
<p align=left>
rb
= <br>
b[0]*fabs(R[1][2]) + b[2]*fabs(R[1][0]);
</p>
<p align=left>
t
= <br>
fabs( T[0]*R[2][1] - <br>
T[2]*R[0][1] );
</p>
<p align=left>
if(
t > ra + rb )<br>
return
false;
</p>
<p align=left>
//L
= A1 x B2<br>
ra
= <br>
a[0]*fabs(R[2][2]) + a[2]*fabs(R[0][2]);
</p>
<p align=left>
rb
= <br>
b[0]*fabs(R[1][1]) + b[1]*fabs(R[1][0]);
</p>
<p align=left>
t
= <br>
fabs( T[0]*R[2][2] - <br>
T[2]*R[0][2] );
</p>
<p align=left>
if(
t > ra + rb )<br>
return
false;
</p>
<p align=left>
//L
= A2 x B0<br>
ra
= <br>
a[0]*fabs(R[1][0]) + a[1]*fabs(R[0][0]);
</p>
<p align=left>
rb
= <br>
b[1]*fabs(R[2][2]) + b[2]*fabs(R[2][1]);
</p>
<p align=left>
t
= <br>
fabs( T[1]*R[0][0] - <br>
T[0]*R[1][0] );
</p>
<p align=left>
if(
t > ra + rb )<br>
return
false;
</p>
<p align=left>
//L
= A2 x B1<br>
ra
= <br>
a[0]*fabs(R[1][1]) + a[1]*fabs(R[0][1]);
</p>
<p align=left>
rb
= <br>
b[0] *fabs(R[2][2]) + b[2]*fabs(R[2][0]);
</p>
<p align=left>
t
= <br>
fabs( T[1]*R[0][1] - <br>
T[0]*R[1][1] );
</p>
<p align=left>
if(
t > ra + rb )<br>
return
false;
</p>
<p align=left>
//L
= A2 x B2<br>
ra
= <br>
a[0]*fabs(R[1][2]) + a[1]*fabs(R[0][2]);
</p>
<p align=left>
rb
= <br>
b[0]*fabs(R[2][1])
+ b[1]*fabs(R[2][0]);
</p>
<p align=left>
t
= <br>
fabs( T[1]*R[0][2] - <br>
T[0]*R[1][2] );
</p>
<p align=left>
if(
t > ra + rb )<br>
return
false;
</p>
<p align=left>
/*no
separating axis found,<br>
the
two boxes overlap */
</p>
<p align=left>