What are the maths behind 3D billboard sprites? (was: 3D transformation matrix to 2D matrix)

Question

I have a 3D point in space. The point\'s exact orientation/position is expressed through a 4x4 transformation matrix.

I want to draw a billboard (3D

Answer 1

if you not using spherical coordinate system then this task is not solvable because discarding Z-coordinate before projection will remove the distance form the projection point and therefore you do not know how to apply perspective.

You have two choices (unless I overlooked something):

1. apply 3D transform matrix

   and then use only x,y - coordinates of the result

2. create 3x3 transformation matrix for rotation/projection

   and add offset vector before or after applying it. Be aware that this approach do not use homogenous coordinates !!!

[Edit1] equations for clarity

Do not forget that 3x3 matrix + vector transforms are not cumulative !!! That is the reason why 4x4 transforms are used instead. Now you can throw away the last row of matrix/vector (Xz,Yz,Zz), (z0) and then the output vector is just (x', y'). Of course after this you cannot use the inverse transform because you lost Z coordinate.

Scaling is done by changing the size of axis direction vectors

Btw. if your projection plane is also XY-plane without rotations then:

x' = (x-x0)*d/(z-z0)
y' = (y-y0)*d/(z-z0)

(x,y,z) - point to project
(x',y') - projected point
(x0,y0,z0) - projection origin
d - focal length

[Edit2] well after question edit the meaning is completely different

I assume you want sprite always facing camera. It is ugly but simplifies things like grass,trees,...

M - your matrix
P - projection matrix inside M
If you have origin of M = (0,0,0) without rotations/scaling/skew then M=P
pnt - point of your billboard (center I assume) (w=1) [GCS]
dx,dy - half sizes of billboard [LCS]
A,B,C,D - projected edges of your billboard [GCS]
[GCS] - global coordinate system
[LCS] - local coordinate system

1. if you know the projection matrix

   I assume it is glFrustrum or gluPerspective ... then:

   (x,y,z,w)=(M*(P^-1))*pnt  // transformed center of billboard without projection
   A=P*(x-dx,y-dy,z,w)
   B=P*(x-dx,y+dy,z,w)
   C=P*(x+dx,y+dy,z,w)
   D=P*(x+dx,y-dy,z,w)
   

2. If your M matrix is too complex for #1 to work

   MM=(M*(P^-1))     // transform matrix without projection
   XX=MM(Xx,Xy,Xz)   // X - axis vector from MM [GCS](look at the image above on the right for positions inside matrix)
   YY=MM(Yx,Yy,Yz)   // Y - axis vector from MM [GCS]
   X =(M^-1)*XX*dx   // X - axis vector from MM [LCS] scaled to dx
   Y =(M^-1)*YY*dy   // Y - axis vector from MM [LCS] scaled to dy
   A = M*(pnt-X-Y)
   B = M*(pnt-X+Y)
   C = M*(pnt+X+Y)
   D = M*(pnt+X-Y)
   

[Edit3] scalling only

MM=(M*(P^-1))      // transform matrix without projection
sx=|MM(Xx,Xy,Xz)|  // size of X - axis vector from MM [GCS] = scale x
sy=|MM(Yx,Yy,Yz)|  // size of Y - axis vector from MM [GCS] = scale y