How to recalculate axis-aligned bounding box after translate/rotate?

Question

When I first load my object I calculate the initial AABB with the max and min (x,y,z) points. But this is in object space and the object moves around the world and more importantly, rotates.

How do I recalculate the new AABB every time the object is translated/rotated? This happens basically every frame, is it going to be a very intensive operation to recalculate the new AABB every frame? If so, what would be the alternative?

I know AABBs will make my collision detection less accurate but it's easier to implement the collision detection code than OBBs and I want to take this one step at a time.

Here's my current code after some insight from the answers below:

typedef struct sAxisAlignedBoundingBox {
    Vector3D bounds[8];
    Vector3D max, min;
} AxisAlignedBoundingBox;

void drawAxisAlignedBoundingBox(AxisAlignedBoundingBox box) {
    glPushAttrib(GL_LIGHTING_BIT | GL_POLYGON_BIT);

    glEnable(GL_COLOR_MATERIAL);
    glDisable(GL_LIGHTING);

    glColor3f(1.0f, 1.0f, 0.0f);

    glBegin(GL_LINE_LOOP);
        glVertex3f(box.bounds[0].x, box.bounds[0].y, box.bounds[0].z);
        glVertex3f(box.bounds[1].x, box.bounds[1].y, box.bounds[1].z);
        glVertex3f(box.bounds[2].x, box.bounds[2].y, box.bounds[2].z);
        glVertex3f(box.bounds[3].x, box.bounds[3].y, box.bounds[3].z);
    glEnd();

    glBegin(GL_LINE_LOOP);
        glVertex3f(box.bounds[4].x, box.bounds[4].y, box.bounds[4].z);
        glVertex3f(box.bounds[5].x, box.bounds[5].y, box.bounds[5].z);
        glVertex3f(box.bounds[6].x, box.bounds[6].y, box.bounds[6].z);
        glVertex3f(box.bounds[7].x, box.bounds[7].y, box.bounds[7].z);
    glEnd();

    glBegin(GL_LINE_LOOP);
        glVertex3f(box.bounds[0].x, box.bounds[0].y, box.bounds[0].z);
        glVertex3f(box.bounds[5].x, box.bounds[5].y, box.bounds[5].z);
        glVertex3f(box.bounds[6].x, box.bounds[6].y, box.bounds[6].z);
        glVertex3f(box.bounds[1].x, box.bounds[1].y, box.bounds[1].z);
    glEnd();

    glBegin(GL_LINE_LOOP);
        glVertex3f(box.bounds[4].x, box.bounds[4].y, box.bounds[4].z);
        glVertex3f(box.bounds[7].x, box.bounds[7].y, box.bounds[7].z);
        glVertex3f(box.bounds[2].x, box.bounds[2].y, box.bounds[2].z);
        glVertex3f(box.bounds[3].x, box.bounds[3].y, box.bounds[3].z);
    glEnd();

    glPopAttrib();
}

void calculateAxisAlignedBoundingBox(GLMmodel *model, float matrix[16]) {
    AxisAlignedBoundingBox box;
    float dimensions[3];

    // This will give me the absolute dimensions of the object
    glmDimensions(model, dimensions);

    // This calculates the max and min points in object space
    box.max.x = dimensions[0] / 2.0f, box.min.x = -1.0f * box.max.x;
    box.max.y = dimensions[1] / 2.0f, box.min.y = -1.0f * box.max.y;
    box.max.z = dimensions[2] / 2.0f, box.min.z = -1.0f * box.max.z;

    // These calculations are probably the culprit but I don't know what I'm doing wrong
    box.max.x = matrix[0] * box.max.x + matrix[4] * box.max.y + matrix[8] * box.max.z + matrix[12];
    box.max.y = matrix[1] * box.max.x + matrix[5] * box.max.y + matrix[9] * box.max.z + matrix[13];
    box.max.z = matrix[2] * box.max.x + matrix[6] * box.max.y + matrix[10] * box.max.z + matrix[14];
    box.min.x = matrix[0] * box.min.x + matrix[4] * box.min.y + matrix[8] * box.min.z + matrix[12];
    box.min.y = matrix[1] * box.min.x + matrix[5] * box.min.y + matrix[9] * box.min.z + matrix[13];
    box.min.z = matrix[2] * box.min.x + matrix[6] * box.min.y + matrix[10] * box.min.z + matrix[14];

    /* NOTE: If I remove the above calculations and do something like this:

             box.max = box.max + objPlayer.position;
             box.min = box.min + objPlayer.position;

             The bounding box will move correctly when I move the player, the same does not
             happen with the calculations above. It makes sense and it's very simple to move
             the box like this. The only problem is when I rotate the player, the box should
             be adapted and increased/decreased in size to properly fit the object as a AABB.
    */

    box.bounds[0] = Vector3D(box.max.x, box.max.y, box.min.z);
    box.bounds[1] = Vector3D(box.min.x, box.max.y, box.min.z);
    box.bounds[2] = Vector3D(box.min.x, box.min.y, box.min.z);
    box.bounds[3] = Vector3D(box.max.x, box.min.y, box.min.z);
    box.bounds[4] = Vector3D(box.max.x, box.min.y, box.max.z);
    box.bounds[5] = Vector3D(box.max.x, box.max.y, box.max.z);
    box.bounds[6] = Vector3D(box.min.x, box.max.y, box.max.z);
    box.bounds[7] = Vector3D(box.min.x, box.min.y, box.max.z);

    // This draw call is for testing porpuses only
    drawAxisAlignedBoundingBox(box);
}

void drawObjectPlayer(void) {
    static float mvMatrix[16];

    if(SceneCamera.GetActiveCameraMode() == CAMERA_MODE_THIRD_PERSON) {
        objPlayer.position = SceneCamera.GetPlayerPosition();
        objPlayer.rotation = SceneCamera.GetRotationAngles();

        objPlayer.position.y += -PLAYER_EYE_HEIGHT + 0.875f;

        /* Only one of the two code blocks below should be active at the same time
           Neither of them is working as expected. The bounding box doesn't is all
           messed up with either code. */

        // Attempt #1
        glPushMatrix();
            glTranslatef(objPlayer.position.x, objPlayer.position.y, objPlayer.position.z);
            glRotatef(objPlayer.rotation.y + 180.0f, 0.0f, 1.0f, 0.0f);
            glCallList(gameDisplayLists.player);
            glGetFloatv(GL_MODELVIEW_MATRIX, mvMatrix);
        glPopMatrix();

        // Attempt #2
        glPushMatrix();
            glLoadIdentity();
            glTranslatef(objPlayer.position.x, objPlayer.position.y, objPlayer.position.z);
            glRotatef(objPlayer.rotation.y + 180.0f, 0.0f, 1.0f, 0.0f);
            glGetFloatv(GL_MODELVIEW_MATRIX, mvMatrix);
        glPopMatrix();

        calculateAxisAlignedBoundingBox(objPlayer.model, mvMatrix);
    }
}

But it doesn't work as it should... What I'm doing wrong?

Answer 1

Simply recompute the AABB of the transformed AABB. This means transforming 8 vertices ( 8 vertex - matrix multiplications ) and 8 vertex-vertex comparisons.

So at initialisation, you compute your AABB in model space : for each x,y,z of each vertex of the model, you check against xmin, xmax, ymin, ymax, ...

Each frame, you generate a new transformation matrix. In OpenGL this is done with glLoadIdentity followed by glTransform/Rotate/Scale (if using the old API). This is the Model Matrix, as lmmilewski said.

You compute this transformation matrix a second time (outside Opengl, for instance using glm). You also can get OpenGL's resulting matrix using glGet.

You multiply each of your AABB's eight vertices by this matrix. Use glm for matrix-vector multiplication. You'll get your transformed AABB (in world space). It it most probably rotated (not axis-aligned anymore)

Now your algorithm probably only work with axis-aligned stuff, hence your question. So now you approximate the new bounding box of the transformed model by takinf the bounding box of the transformed bounding box:

for each x,y,z of each vertex of the new AABB, you check against xmin, xmax, ymin, ymax, ... this gives you an world-space AABB that you can use in your clipping algorithm.

This is not optimal (AABB-wise), you'll get lots of empty space, but performance-wise, it's much much better that recomputing the AABB of the whole mesh.

---

As for the transformation matrix, in drawObjectPlayer:

        gLLoadIdentity();
        glTranslatef(objPlayer.position.x, objPlayer.position.y, objPlayer.position.z);
        glRotatef(objPlayer.rotation.y + 180.0f, 0.0f, 1.0f, 0.0f);
        glGetFloatv(GL_MODELVIEW_MATRIX, mvMatrix);
  // Now you've got your OWN Model Matrix (don't trust the GL_MODELVIEW_MATRIX flag : this is a workaround, and I know what I'm doing ^^ )

        gLLoadIdentity(); // Reset the matrix so that you won't make the transformations twice
        gluLookAt( whatever you wrote here earlier )
        glTranslatef(objPlayer.position.x, objPlayer.position.y, objPlayer.position.z);
        glRotatef(objPlayer.rotation.y + 180.0f, 0.0f, 1.0f, 0.0f);
      // Now OpenGL is happy, he's got his MODELVIEW matrix correct ( gluLookAt is the VIEW part; Translate/Rotate is the MODEL part
        glCallList(gameDisplayLists.player); // Transformed correcty

Can't explain further than that... as said in the comments, you had to do it twice. You wouldn't have these problems and ugly workarounds in OpenGL 3, btw, because you'd be fully responsible of your own matrices. Equivalent in OpenGL 2 :

glm::mat4 ViewMatrix = glm::LookAt(...);
glm::mat4 ModelMatrix = glm::rotate() * glm::translate(...);
// Use ModelMatrix for whatever you want
glm::mat4 ModelViewMatrix = ViewMatrix * ModelMatrix;
glLoadMatrix4fv( &ModelViewMatrix[0][0] ); // In opengl3 you would use an uniform instead

much cleaner right

Answer 2

To quote a previous response: AABB @ Stackoverflow

"Sadly yes, if your character rotates you need to recalculate your AABB . . .

Skurmedel

---

The respondent's suggestion, and mine, is to implement oriented bounding boxes once you have AABB working, and also to note you can make aabb's of portions of a mesh to fudge collision detection with greater accuracy than 1 enormous box for each object.

Answer 3

To do that you have to loop over every vertex, calculate it's position in the world (multiply by modelview) and find minimum / maximum vertex coordinates within every object (just like when you compute it for the first time).

You can scale a bit your AABB so that you don't have to recalculate it - it is enough to enlarge it by factor sqrt(2) - your rotated object then always fits in AABB.

There is also a quesion in which direction you rotate? If always in one then you can enlarge AABB only in that direction.

Optionally you can use bounding spheres instead of AABBs.Then you don't care about rotation and scaling is not a problem.

At the end I must ask if you are sure that this is a bottleneck in your application. I believe it's not and in that case I would use first option I mentioned (iterate over all vertices).

Answer 4

Yep, you can transform the 8 corner vertices and do min/max on the results, but there is faster way, as described by Jim Arvo from his chapter in Graphics Gems (1990).

Performance-wise, Arvo's method is roughly equivalent to 2 transforms instead of 8 and basically goes as follows (this transforms box A into box B)

split the transform into a translation vector (T) and a 3x3 rotation (M).
B = zero-volume AABB at T
for each element (i,j) of M:
   a = M[i][j] * A.min[j]
   b = M[i][j] * A.max[j]
   B.min[i] += a < b ? a : b
   B.max[i] += a < b ? b : a
return B

One variation of Arvo's method uses center / extent representation rather than mix / max, which is described by Christer Ericson in Real-Time Collision Detection (photo).

Complete C code for Graphics Gems article can be found here.

Answer 5

Why not use your GPU? Today I implimented a solution of this problem by rendening a couple of frames.

1. Temporary place your camera over the object, above it, pointing down at the object.
2. Render only your object, with out lights or anything.
3. Use orthographic projection too.
4. Then read the frame buffer. Rows and columns of black pixels means the model isn't there. Hit a white pixel - you hit one of the model AABB borders.

I know this isn't a solution for all the cases, but with some prior knowledge, this is very efficient.

For rendering off screen see here.