104 posts
SAT Collision detection and response
Edited by C_Worm on Reason: Initial post
Hey I managed to implement the serparated axis theorem(SAT), however im finding it very hard to understand how to extract the so called "minimum translation vector" so that the colliding polygons can be separated.

Any advice on how to implement this would be appreciated.

Here's my Collision function:

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 struct CollisionInfo { bool collision; V2f minVec; }; CollisionInfo Collision_Entity_SAT(Entity e1, Entity e2) { CollisionInfo result = {}; // get normal of both sides V2f normal1 = Normal_V2f(e1.pTop, e1.pRight); V2f normal2 = Normal_V2f(e1.pTop, e1.pLeft); float dp1[4] = {}; float dp2[4] = {}; float dp3[4] = {}; float dp4[4] = {}; // get dotproduct of each entities corner against both normals dp2[0] = DotProduct_V2f(normal1, e1.pTop); dp2[1] = DotProduct_V2f(normal1, e1.pLeft); dp2[2] = DotProduct_V2f(normal1, e1.pRight); dp2[3] = DotProduct_V2f(normal1, e1.pBottom); dp1[0] = DotProduct_V2f(normal1, e2.pTop); dp1[1] = DotProduct_V2f(normal1, e2.pLeft); dp1[2] = DotProduct_V2f(normal1, e2.pRight); dp1[3] = DotProduct_V2f(normal1, e2.pBottom); dp4[0] = DotProduct_V2f(normal2, e1.pTop); dp4[1] = DotProduct_V2f(normal2, e1.pLeft); dp4[2] = DotProduct_V2f(normal2, e1.pRight); dp4[3] = DotProduct_V2f(normal2, e1.pBottom); dp3[0] = DotProduct_V2f(normal2, e2.pTop); dp3[1] = DotProduct_V2f(normal2, e2.pLeft); dp3[2] = DotProduct_V2f(normal2, e2.pRight); dp3[3] = DotProduct_V2f(normal2, e2.pBottom); float max_dp1 = Max_f(dp1, sizeof(dp1) / sizeof(float)); float min_dp1 = Min_f(dp1, sizeof(dp1) / sizeof(float)); float max_dp2 = Max_f(dp2, sizeof(dp2) / sizeof(float)); float min_dp2 = Min_f(dp2, sizeof(dp2) / sizeof(float)); float max_dp3 = Max_f(dp3, sizeof(dp3) / sizeof(float)); float min_dp3 = Min_f(dp3, sizeof(dp3) / sizeof(float)); float max_dp4 = Max_f(dp4, sizeof(dp4) / sizeof(float)); float min_dp4 = Min_f(dp4, sizeof(dp4) / sizeof(float)); V2f minVec = {}; if(max_dp1 > min_dp2 && min_dp1 < max_dp2 && max_dp3 > min_dp4 && min_dp3 < max_dp4) { result.collision = true; } else { result.collision = false; } return result; } 
19 posts
SAT Collision detection and response
Note: I hesitated to reply to this because its been quite a while since I implemented SAT (or any collision detection/response/physics for that matter) so take this with your favourite amount of salt, its basically an educated guess based on what I remember and can reason out. I happily defer to anybody who is more confident on this topic. Hopefully this helps though :)

When using SAT you have a collection of axes that you're projecting your polygons onto and checking if there are any on which they don't overlap. If they overlap on all axes then the polygons themselves overlap. However as part of this calculation, you already compute (or can easily compute) the *distance* by which they overlap on each axis. The axis with the lowest such distance is the one in which the polygons are conceptually "the closest to being separated".

If you move the polygons apart on that axis then you will have an axis along which they are now separated and therefore can be sure that you have resolved the collision! As to whether this is the *minimum* translation to separate them, I suspect so based on the hand-waving line of reasoning that if you followed this procedure for any axis you could always separate the polygons by moving them some distance but since we picked the axis along which the polygons overlap the least, we know that the distance we need to move them is minimal among all the axes we're checking.

So now we have an axis and a distance to move in order to separate our polygons. You may want to just move one of them the full distance, you may want to move them both half the distance (but in opposite directions). It depends on the objects and the behaviour you're looking for. This is definitely not something I have researched/implemented myself but I believe you tend to run into problems with this sort of thing when you have two or more pairs of objects you're checking for collisions between, where in resolving one intersection you create another (imagine a ball moving into the corner where two walls meet, or getting squashed between two rectangles getting closer together). You'll need some more involved machinery to resolve that with any amount of reliability.

Also note that you're only checking two axes, which is sufficient only if your objects are all axis-aligned rectangles. If you ever rotate anything you'll need to check the other 2 normals as well. :)
104 posts
SAT Collision detection and response
Thanks, nice to "hear" your voice on this!
104 posts
SAT Collision detection and response
Edited by C_Worm on
So I've changed the code and now im checking all the axis on both entities however the resolving of the collision works on TWO sides only, when the moving object collides on the "upper" or the "right" side of the other object the moving object just pass through it and instantly ends up on the other side of it.

Im checking a isometric tile against another isometric tile and a simple axis alligned square.

I've also tested all my vector/math functions and the seem to work.

I'll paste the SAT function and the place in the code where i use it :)

SAT_Function()

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 CollisionInfo Collision_Entity_SAT(Entity e1, Entity e2) { CollisionInfo result = {}; float dot_e1[4] = {}; float dot_e2[4] = {}; V2f normals_e1[4] = {}; V2f normals_e2[4] = {}; V2f mtv = {}; float minDot_e1 = 0.0f; float maxDot_e1 = 0.0f; float minDot_e2 = 0.0f; float maxDot_e2 = 0.0f; float overlap = 100000000.0f; float tempOverlap = 0.0f; // Get normals_e1 of all sides and make them unit vectors for(i32 i = 0; i < 4; i++) { if(i == 3) { normals_e1[i] = Unit_V2f(Normal_V2f(e1.points[i], e1.points[(i - 3)])); } else { normals_e1[i] = Unit_V2f(Normal_V2f(e1.points[i], e1.points[(i + 1)])); } } // Get normals_e2 of all sides and make them unit vectors for(i32 i = 0; i < 4; i++) { if(i == 3) { normals_e2[i] = Unit_V2f(Normal_V2f(e2.points[i], e2.points[(i - 3)])); } else { normals_e2[i] = Unit_V2f(Normal_V2f(e2.points[i], e2.points[(i + 1)])); } } for(i32 j = 0, i = 0; j < 4; j++) { for(i = 0; i < 4; i++) { // calc dot products of all points one normal at a time dot_e1[i] = DotProduct_V2f(normals_e1[j], e1.points[i]); dot_e2[i] = DotProduct_V2f(normals_e1[j], e2.points[i]); } // Get min/max Dotproducts of each entity against 1 normal at a time minDot_e1 = Min_f(dot_e1, sizeof(dot_e1)/sizeof(float)); maxDot_e1 = Max_f(dot_e1, sizeof(dot_e1)/sizeof(float)); minDot_e2 = Min_f(dot_e2, sizeof(dot_e2)/sizeof(float)); maxDot_e2 = Max_f(dot_e2, sizeof(dot_e2)/sizeof(float)); // Check for overlapp tempOverlap = OverLap(minDot_e1, maxDot_e1, minDot_e2, maxDot_e2); // No overlapp = no collision if(tempOverlap <= 0.0f) { result.collision = false; result.mtv = v2f(0.0f, 0.0f); result.minPenetration = 0.0f; return result; } // some overlapp occured = collision on the axis currently under control else { if(tempOverlap < overlap) { overlap = tempOverlap; mtv = normals_e1[j]; result.minPenetration = overlap; } } } // Do it all again but check against the normals on the other entity for(i32 j = 0, i = 0; j < 4; j++) { for(i = 0; i < 4; i++) { // calc dot products of all points one normal at a time dot_e1[i] = DotProduct_V2f(normals_e2[j], e1.points[i]); dot_e2[i] = DotProduct_V2f(normals_e2[j], e2.points[i]); } // get min/max DP's minDot_e1 = Min_f(dot_e1, sizeof(dot_e1)/sizeof(float)); maxDot_e1 = Max_f(dot_e1, sizeof(dot_e1)/sizeof(float)); minDot_e2 = Min_f(dot_e2, sizeof(dot_e2)/sizeof(float)); maxDot_e2 = Max_f(dot_e2, sizeof(dot_e2)/sizeof(float)); // Check for overlapp tempOverlap = OverLap(minDot_e1, maxDot_e1, minDot_e2, maxDot_e2); if(tempOverlap <= 0.0f) { result.collision = false; result.mtv = v2f(0.0f, 0.0f); result.minPenetration = 0.0f; return result; } else { if(tempOverlap < overlap) { overlap = tempOverlap; mtv = normals_e2[j]; result.minPenetration = overlap; } } } // Collision occured and we obtain the minimum penetration magnitude and the corresponding axis result.mtv = Scalar_V2f(result.minPenetration, mtv); result.collision = true; return result; } 

where I use it

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35  CollisionInfo ci_0 = {}; CollisionInfo ci_1 = {}; // COLLISION CHECKING ci_1 = Collision_Entity_SAT(e[tile_00], e[tile_00 + 1]); ci_0 = Collision_Entity_SAT(e[tile_00], e[ghostSorc]); if(ci_1.collision || ci_0.collision) { if(ci_1.collision) { printf("c_1.overlap: %f\n", ci_1.minPenetration); printf("c_1.mtv: ( %f, %f )\n\n", ci_1.mtv.x, ci_1.mtv.y); e[tile_00].SetSolidColor(0.0f, 0.5f, 0.0f, 1.0f); e[tile_00].Move(ci_1.mtv); } if(ci_0.collision) { printf("c_0.overlap: %f\n", ci_0.minPenetration); printf("c_0.mtv: ( %f, %f )\n\n", ci_0.mtv.x, ci_0.mtv.y); e[tile_00].SetSolidColor(0.0f, 0.5f, 0.0f, 1.0f); e[tile_00].Move(ci_0.mtv); } } else { e[tile_00].SetSolidColor(0.2f, 0.0f, 0.2f, 1.0f); } 
Simon Anciaux
1275 posts
SAT Collision detection and response
Edited by Simon Anciaux on
What does the OverLap function does (what is the code) ?

Have you tried stepping in the code to see what is breaking ?

Have you tried drawing the axis, the projections and the result to see if they look correct ?
104 posts
SAT Collision detection and response
Yes i've stepped in and looked at it and from what i can see the axis are correct.

The projections (Dot products) also seems right.

.... been working on this collision-system for like atleast 2 weeks now xD

Overlap():
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 float OverLap(float min1, float max1, float min2, float max2) { float min = 0.0f; float max = 0.0f; float overlap = 0.0f; if(max1 < max2) { max = max1; } else { max = max2; } if(min1 < min2) { min = min2; } else { min = min1; } overlap = max - min; return overlap; } 
Simon Anciaux
1275 posts
SAT Collision detection and response
Edited by Simon Anciaux on Reason: fixed code
I did a small test out of curiosity, and it seems that the result of my code (which I think should be similar to yours) sometimes gives a result meant to move shape A and other times to move shape B. That means, that the code gives the minimum amount you have to move, and the axis, but not the direction.

I fixed it by comparing minDot_e1 and minDot_e2 and returning a negative direction (relative to the axis) if minDot_e1 < minDot_e2. Meaning the vector to move a in sat( a, b ) is axis * magnitude * direction. I think it should do it but I didn't test much as my "setup" is no practical to do so.

The following code will produce a series of images, with the two shapes in white, and a red line indicating in which direction moving the first shape to solve the collision (the first shape is the one not moving, at the center). You can play with the values to test different shapes, movements...

If you comment
 1 2 3 if ( a_min < b_min ) { result.direction = -1; } 

you can see the incorrect result when shape B has nearly exited shape A to the bottom left (the red arrow is in the wrong direction).

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 #include #include typedef float r32; typedef unsigned int u32; typedef struct vec2_t { r32 x, y; } vec2_t; r32 v2_dot( vec2_t a, vec2_t b ) { r32 result = a.x * b.x + a.y * b.y; return result; } vec2_t v2_perp( vec2_t a ) { vec2_t result; result.x = -a.y; result.y = a.x; return result; } vec2_t v2_normalized( vec2_t a ) { r32 length = sqrtf( a.x * a.x + a.y * a.y ); a.x = a.x / length; a.y = a.y / length; return a; } #define min( a, b ) ( ( ( a ) < ( b ) ) ? ( a ) : ( b ) ) #define max( a, b ) ( ( ( a ) > ( b ) ) ? ( a ) : ( b ) ) #define swap( a, b, type ) { type __backup__ = ( a ); ( a ) = ( b ); ( b ) = __backup__; } typedef struct sat_result_t { vec2_t axis; r32 magnitude; r32 direction; } sat_result_t; sat_result_t sat( vec2_t* a, vec2_t* b ) { sat_result_t result = { 0 }; result.magnitude = 1000000.0f; int stop = 0; vec2_t axis[ 8 ]; for ( int i = 0; i < 4; i++ ) { vec2_t va = { a[ i + 1 ].x - a[ i ].x, a[ i + 1 ].y - a[ i ].y }; vec2_t vb = { b[ i + 1 ].x - b[ i ].x, b[ i + 1 ].y - b[ i ].y }; axis[ i ] = v2_normalized( v2_perp( va ) ); axis[ 4 + i ] = v2_normalized( v2_perp( vb ) ); } for ( int axis_index = 0; axis_index < 8; axis_index++ ) { r32 a_min = 1000000.0f; r32 a_max = -a_min; r32 b_min = 1000000.0f; r32 b_max = -b_min; for ( int point_index = 0; point_index < 4; point_index++ ) { r32 a_dot = v2_dot( axis[ axis_index ], a[ point_index ] ); r32 b_dot = v2_dot( axis[ axis_index ], b[ point_index ] ); a_min = min( a_min, a_dot ); a_max = max( a_max, a_dot ); b_min = min( b_min, b_dot ); b_max = max( b_max, b_dot ); } r32 overlap_max = min( a_max, b_max ); r32 overlap_min = max( a_min, b_min ); r32 overlap = overlap_max - overlap_min; if ( overlap <= 0 ) { result.axis.x = 0; result.axis.y = 0; result.magnitude = 0; result.direction = 0; break; } else if ( overlap < result.magnitude ) { result.axis = axis[ axis_index ]; result.magnitude = overlap; result.direction = 1; if ( a_min < b_min ) { result.direciton = -1; } } } return result; } #define width 512 #define height 512 u32 buffer[ width * height * 4 ] = { 0 }; void draw_line( vec2_t p1, vec2_t p2, u32 color ) { r32 dx = fabsf( p2.x - p1.x ); r32 dy = fabsf( p2.y - p1.y ); if ( dx >= dy ) { if ( p1.x > p2.x ) { swap( p1, p2, vec2_t ); } int sign = 1; if ( p1.y > p2.y ) { sign = -1; } r32 increment = dy / dx; u32 x = ( u32 ) p1.x; u32 x_end = ( u32 ) p2.x; u32 y = ( u32 ) p1.y; r32 error = 0.5f; while ( x <= x_end ) { buffer[ y * width + x ] = color; error += increment; if ( error >= 1.0f ) { y = ( u32 ) ( y + sign ); error -= 1.0f; } x++; } } else { if ( p1.y > p2.y ) { swap( p1, p2, vec2_t ); } int sign = 1; if ( p1.x > p2.x ) { sign = -1; } r32 increment = dx / dy; u32 y = ( u32 ) p1.y; u32 y_end = ( u32 ) p2.y; u32 x = ( u32 ) p1.x; r32 error = 0.5f; while ( y <= y_end ) { buffer[ y * width + x ] = color; error += increment; if ( error >= 1.0f ) { x = ( u32 ) ( x + sign ); error -= 1.0f; } y++; } } } int main( int argc, char** argv ) { unsigned char tga_header[ 18 ] = { 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0x2, 0, 0x2, 32, 8, }; vec2_t a_pos = { 255.0f, 255.0f }; vec2_t b_pos = { 350.0f, 290.0f }; for ( int step = 0; step < 150; step++ ) { vec2_t a[ 5 ] = { a_pos.x + 0.0f, a_pos.y - 50.0f, a_pos.x + 50.0f, a_pos.y + 0.0f, a_pos.x + 0.0f, a_pos.y + 50.0f, a_pos.x + -50.0f, a_pos.y + 0.0f, }; a[ 4 ] = a[ 0 ]; vec2_t b[ 5 ] = { b_pos.x + 0.0f, b_pos.y - 50.0f, b_pos.x + 50.0f, b_pos.y + 0.0f, b_pos.x + 0.0f, b_pos.y + 50.0f, b_pos.x + -50.0f, b_pos.y + 0.0f, }; b[ 4 ] = b[ 0 ]; sat_result_t result = sat( a, b ); for ( int i = 0; i < width * height; i++ ) { buffer[ i ] = 0xff000000; } for ( int i = 0; i < 4; i++ ) { draw_line( a[ i ], a[ i + 1 ], 0xffffffff ); draw_line( b[ i ], b[ i + 1 ], 0xffffffff ); } if ( result.magnitude ) { vec2_t start = { 255.0f, 255.0f }; vec2_t end = { 255.0f + result.axis.x * result.magnitude * result.direction, 255.0f + result.axis.y * result.magnitude * result.direction }; draw_line( start, end, 0xffff0000 ); } char filename[ ] = "render_000.tga"; char c = ( char ) ( step / 100 ); char d = ( char ) ( ( step - ( c * 100 ) ) / 10 ); char u = ( char ) ( step - c * 100 - d * 10 ); filename[ 7 ] = '0' + c; filename[ 8 ] = '0' + d; filename[ 9 ] = '0' + u; FILE* file = fopen( filename, "wb" ); fwrite( tga_header, 18, 1, file ); fwrite( buffer, 512 * 512 * 4, 1, file ); fclose( file ); b_pos.x -= 1; b_pos.y -= 1; } return 0; } 
104 posts
SAT Collision detection and response
Just wow, how can you be so good?! xD

Thanks so much, that was the tiny detail that was missing, so thankful for your response :D

Now i've got the sat collision implemented, and from here on i guess its all about experimenting with difference responses! =)
Simon Anciaux
1275 posts
SAT Collision detection and response
Once I had a visualization, it quickly became obvious what was the issue. Taking some time to create (good) visualization is always worth it.