Math Question, I suck at math

[NexusOne]

Ok, I have a vector of length 1, and what I want to do is calculate the rotation matrix that would move the point (0, 1, 0) to the end of the vector. How do I do this?

Here's what I've been trying, it may be wrong:

double xAngle = 0, yAngle = 0, zAngle = 0;
if(vector.y == 0) xAngle = 0;
else xAngle = Math.atan(vector.z / vector.y);
if(vector.z == 0) yAngle = 0;
else yAngle = Math.atan(vector.x / vector.z);
if(vector.y == 0) zAngle = 0;
else zAngle = Math.atan(-1 * vector.x / vector.y);

pointX = 0;
pointY = 1;
pointZ = 0;

//hardcoded calculation that would be done in a rotational matrix
pointY = Math.cos(xAngle)*pointY + Math.sin(xAngle)*pointZ;
pointZ = -1*Math.sin(xAngle)*pointY + Math.cos(xAngle)*pointZ;
pointX = Math.cos(yAngle)*pointX + -1*Math.sin(yAngle)*pointZ;
pointZ = Math.sin(yAngle)*pointX + Math.cos(yAngle)*pointZ;
pointX = Math.cos(zAngle)*pointX + Math.sin(zAngle)*pointY;
pointY = -1*Math.sin(zAngle)*pointX + Math.cos(zAngle)*pointY;


If there exists some method which can do this for me, or some short vector or matrix manipulation (I have a vecmath.jar with simple vector and matrix classes and methods), or you have some short code that does just what i want, please tell me. Thanks a lot!

[crystalsquid]

This is easier than that. Matrices are just arrays of axis vectors. What you are after is a matrix that has the y-axis pointing along the vector you describe. This pseudo-code should show how to do it, it will need tweaking to whichever vactor & matrix classes you use:

1) Start with our desired y-axis:

 myVector y = vector;

Then set the z-axis to point up Z:

 myVector z(0, 0, 1);

You then build the true x-axis by:

 myVector x = y.crossProduct( z );
 x.normalize();

Finally, build the true z-axis:

 z = x.crossProduct( y );

Then build the matrix:

 myMatrixClass m;
 m.setXAxis( x );
 m.setYAxis( y );
 m.setZAxis( z );

And that is your rotation matrix.
Note that this one should leave +z in roughly the same direction as the original z-axis, so the point (0,0,1) should come out with a positive z always. If you set x instead of z and change the cross-products to:
z = x.cross( y ); x.normalize(); x = y.cross( z );
That will keep the x-axis in the same direction instead.
Hope this helps,

Dom

[swpalmer]

Do you need that step of building the 'true' Z-axis? Since you used the 0,0,1 Z to create X from Y&Z in the first place, shouldn't it come out the same (0,0,1) when you derive Z from X&Y?

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[crystalsquid]

Yep, because:

(y-axis fixed)
z-axis setto up <- this is probably not 90 degrees to Y

make x-axis - due to cross product, this IS perpendicular to the y and z axis, but z is still not perpendicular to Y

build true Z - this one will now be truly perpendicular.

The standard use for this is aligning a game camera in the world (except z is fixed, and y is up). You make a normal from the camera position to the point of interest (character usually), and set the camera z-axis to this (or away from in OpenGL I think). You then set Y axis to straight up. The first cross product gets you the right axis for the camera. The last cross product gets you the 'tilted' up-axis of the camera (for example if you were looking 45 degrees down along the z axis, the y-axis should come out as (0, 0.707, 0.707) not the (0,1,0) we fed in at the start.
Hope that makes sense (I have a tendancy to ramble...)

- Dom

[swpalmer]

makes perfect sense.  I don't know where my brain was.  It's been far to long since I've done some real math...  I keep meaning to find the time to do some 3D stuff to refresh my memory but it just hasn't happned yet... Got any spare time you can lend me?

One note:
I think this will fail if the desired y is such that it also just happens to be pointing straight 'up', since ZY won't define a plane.  Be careful of those edge conditions that cause errors when you least expect them.