Matrix to transform vector MA to MB same length

OpenGL
1

Hello,
i am working on a 3D project. some where in my program i need to draw a plan that is perpendicular to a vector MB. for doing this i need to know the (x,y,z) of each voxel in this plan (a portion of the plan or window).
for this i need to make a marix that transform a well known vector MA (for exemple parallel to X axis) to a known vector MB of the same length. why ? because I will calculate the (x,y,z) of the plan perpendicular to X axis (witch is easy to calculate) and then multiplay by the matrix to obtain the (x,y,z) in the new plan (perpendicular to MB)
So i need a rotation matrix (because of the same origin M of the two vectors). I think i can do only two rotations (on Y and Z axis) to obtain the new vector MB.
In the literature I have only found rotation about x, y or z axis or even rotation about a vector v. but i can t find a matrix to have a vector from an other.
I don t know if i can consider the multiplication of two matrix (one alpha rotation about y and beta rotation about z) and try to find alpha and beta.
excuse me but i am not good at maths
think you

2

To find a matrix that transform a vector MA to a vector MB you can try this:

  1. If MA and MB are not unit length, normalize it.

  2. Calculate the dot product of MA and MB, this will give the cosinus of the angle between MA and MB. To get the angle, take the arccos of the result.

  3. Calculate the cross product of MA and MB, this will give a vector for the axe of rotation.

  4. Now we have an angle and a vector. Then you can use the Rodrigues formula to calculate a rotation matrix to rotate MA to MB.

hope that help

3

perfect think you

4

Take care of the cross product because you will have to normalize it. And there are two special case when the cross product is zero :

  1. MA = MB in this case the rotation matrix is identity.

  2. MA = -MB, in this case you will have to choose a vector that lie
    on the plane (MA_x,MA_y,MA_z).

5

ok think you

6

Excuse me, but I still have one small problem.
the dot product will give me cos(teta) so the angle my be a or -a !!
and the cross product give me |sin(teta)| (absolute value). so how can I know if it is a or -a ??
think you

7

The simple thing to do is to try both.

if MA * Rodrigues(a,vect) = MB then the angle is a
if MA * Rodrigues(-a,vect) = MB then the angle is -a

If the dot product is 0, the possible angles
are PI/2 and -PI/2.

The compare function between two vectors is something like this to avoid floating point precision problem.


bool compare(vect1,vect2)
{
 EPSILON = 0.001;
 if( (fabs(vect1.x - vect2.x) < EPSILON) && (repeat for y and z))
 {
  return(true);
 }
 else
{
 return(false);
}
}
8

I think there is a clever way to determine the sign of angle a.

Take the determinant of the two vector

| 1 1 1 |
| MB_X MB_Y MB_Z |
| MA_X MA_Y MA_Z |

if the determinant of this matrix is positive,
the position of MB is clockwise relative of MA and the angle a
is positive.

if the determinant is negative, the position of MB is counter-clockwise relative of MA and the angle a
is negative.

9

The (anticommutative) crossproduct and the windedness of the rotation around that crossproduct (which you use as the axis fpr rotation) will incorporate that information. ( a x b = - b x a )
So you have the same Angle if you swap two vectors, but in the same step you are negating the rotation axis and thus the orientation of the rotation.