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# aligning two vectors

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### #1davidko  Members   -  Reputation: 122

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Posted 29 September 2001 - 12:55 PM

Does anyone know of source code that generates a rotation matrix to align two vectors? I have written the code, but for some reason, the vectors do not match up exactly. It might be floating point errors, but the error can''t be THAT bad. Thank you.

### #2Sneftel  Senior Moderators   -  Reputation: 1788

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Posted 29 September 2001 - 07:15 PM

to generate this matrix, simply calculate the cross-product of the two vectors, calculate the angle between them, and make a standard rotation matrix with the cross-product as the axis, and the angle as... the angle.

A rotation matrix around an arbitrary axis may be found at http://www.gamedev.net/reference/articles/article1199.asp ... this one is, in turn, taken from "Graphics Gems".

If the vectors do not line up afterwards, this may be because you are rotating in the wrong direction.

Edited by - sneftel on September 30, 2001 2:17:11 AM

### #3davidko  Members   -  Reputation: 122

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Posted 29 September 2001 - 07:43 PM

Okay, I will try that. I was just trying to align the vector with the z-axis. The way I was doing it before, I was projecting the vector to two planes (XZ and XY) and calculating the angle with which to rotate about the x and y axis. I used those angles to get two rotation matricies and concatenated them. Unfortunately, the vectors weren''t matching up exactly. It was close, but definitely not close enough....

### #4Succinct  Members   -  Reputation: 169

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Posted 03 October 2001 - 11:19 AM

Yes, I''ve worked on things like this.

As Sneftel said, find the cross product of the two vectors and then their dot product.

The result of the cross product is the axis on which to rotate the vector, and the dot product is the cosine of the angle.

pseudocode would be:
  Vector c = Cross( v,Vector( 0,0,1 ) );float d = Dot( v,Vector( 0,0,1 ) );float Angle = acos( d );Matrix R = RotationMatrix( Angle,Axis );

You can optimize the crosses and dots because you always know that the z axis is (0,0,1):
  Vector Axis = Vector( v.y,-v.x,0 )float Angle = v.z

To generate a Rotation matrix from an angle/axis:
  Matrix S = ZeroMatrix;S[0][1] = -Axis.z;S[0][2] = Axis.y;S[1][0] = Axis.z;S[1][2] = -Axis.x;S[2][0] = -Axis.y;S[2][1] = Axis.x;R = IdentityMatrix + S*sin( Angle ) + (S*S)*(1 - cos( Angle ));

Hope that helps!

### #5davidko  Members   -  Reputation: 122

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Posted 03 October 2001 - 05:00 PM

Thanks for link and the code! I haven''t tried it out, but I will soon!

### #6Dobbs  Members   -  Reputation: 164

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Posted 04 October 2001 - 06:40 AM

One thing that may affect this is that when the two vectors are roughly 180 degrees apart the axis of rotation that you generate from the cross product probably won''t be terribly accurate because of numerical error. Just something to think about.

### #7Thrump  Members   -  Reputation: 169

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Posted 04 October 2001 - 11:07 AM

I did something like this a long time ago. The way I did it, Matrices for 180 degree vectors were way off. Actually, seemed pretty much random.

### #8davidko  Members   -  Reputation: 122

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Posted 04 October 2001 - 07:53 PM

I think the greatest accuracy comes when two normals are perpendicular.

Probably a robust formulation of this algorithm would be to rotate the normal to an intermediary vector, and then to the final destination. The two matricies can then be concatenated. I suppose you could do this when the vectors are very close to parallel, although I suspect you'd lose some precision this way...

Edited by - davidko on October 5, 2001 2:54:35 AM

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