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Orthonormalize Two Vectors


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#1 Steve_Segreto   Crossbones+   -  Reputation: 1470

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Posted 17 October 2010 - 08:44 AM

I'm trying to implement something like Unity's Vector3.Orthonormalize() function.

Vector3.OrthoNormalize

Since the documentation for this function states that both vectors are normalized and then the second vector is made to be orthogonal to the first (i.e. 90 degrees angle between second and first vector), my question is whether I should even bother with G-S Orthonormalization or just use this:


Vector3 OrthoNormalize( Vector 3 normal, Vector3 tangent )
{
normal.normalize();
tangent.normalize();

return tangent.cross( normal );
}




So is the cross-product of two normalized vectors a third vector that has a 90 degree angle to the first one?

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#2 macnihilist   Members   -  Reputation: 368

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Posted 17 October 2010 - 09:18 AM

Quote:

So is the cross-product of two normalized vectors a third vector that has a 90 degree angle to the first one?

Yes, but in general it won't have a magnitude of one. The magnitude of the cross product is the area of the parallelogram the two arguments form. And if these two vectors have unit length but are not perpendicular the magnitude will be less than one.
Also, I would let the function take the both tangents by reference, since the first one may have to be modified to get an orthonormal basis (and the second one may have to be constructed). Or, which I usually do, let it operate on a matrix.

EDIT:
To answer the first question: As far as I know you'd have to use either GS or two cross products + normalizations:

n.normalize();
v = cross(n,u);
v.normalize();
u = cross(v,n);



[Edited by - macnihilist on October 17, 2010 3:18:04 PM]

#3 Steve_Segreto   Crossbones+   -  Reputation: 1470

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Posted 17 October 2010 - 01:48 PM

How about this? Would this work to orthonormalize the tangent vector to the normal vector?


void D3DXVec3OrthoNormalize( D3DXVECTOR3 *normal, D3DXVECTOR3 *tangent )
{
D3DXVec3Normalize( normal, normal );
D3DXVec3Normalize( tangent, tangent );

D3DXVec3Cross( tangent, normal, tangent );
}




#4 apatriarca   Crossbones+   -  Reputation: 1604

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Posted 17 October 2010 - 02:25 PM

Isn't it exactly the same logic as the code in your first post? Both your new code and the first don't work as already explained by macnihilist. I don't understand why you don't want to use GS ortho-normalization since it is cheaper and only slightly longer to write:

void D3DXVec3OrthoNormalize( D3DXVECTOR3 *normal, D3DXVECTOR3 *tangent )
{
D3DXVec3Normalize( normal, normal );

D3DXVECTOR3 proj;
D3DXVec3Scale( &proj, normal, D3DXVec3Dot( tangent, normal) );

D3DXVec3Subtract( tangent, tangent, proj );

D3DXVec3Normalize( tangent, tangent );
}





#5 Steve_Segreto   Crossbones+   -  Reputation: 1470

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Posted 17 October 2010 - 04:22 PM

Thank you macnihilist and apatriarca!

Rating both of you up. Sorry, my brain was working a bit slow ;)

#6 Petry   Members   -  Reputation: 100

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Posted 12 November 2010 - 06:00 AM

Hey so, I found this and was thinking that I can help you if you can help me.

When you cross two orthonomal vectors together, you will always get another orthonormal vector (try crossing (1,0,0) and (0,1,0), which yields (0,0,1)).

Also note, that whenever a cross is performed you will always return a vector that is orthogonal (i.e. 90 degrees in between the two)to BOTH of the vectors that you crossed together.

You can prove this by doting the returned vector with the two input vectors, which will yield a 0 (Dot(a,b)=||a|| *||b|| Cos(angle between the two)) and since Cos(90) is 0, thus proved.

Sometimes you will get the 0 vector, this is technically orthogonal to every vector out there, which means that the two vectors you crossed together were similar (pointing in the same direction).

But as far as what I was wondering, I have been using this stuff in Physics, Statics, Calculus, etc and have been wondering how it can be applied to actual problems, so if you don't mind explaining why you were wanting to do this, that would be awesome :).

I have programmed for about a year and a half so I don't mind if you use programming terminoligy to explain.

Thanks :)

Petry

#7 ma_hty   Members   -  Reputation: 100

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Posted 14 November 2010 - 02:45 AM

Quote:
Original post by Petry
... why you were wanting to do this, ...


There are a number of 3D rendering algorithms that depend on local coordinates system explicitly, e.g. dynamic bump mapping and anisotropic lighting (e.g. WARD lighting model).

For a 3D model, the normals and the tangents are defined at vertices and are interpolated for the pixel operations. Although vertex normal and vertex tangent are orthonormal to each other, the interpolated normal and tangent are not orthonormal in general. So, when we try to form a local coordinates system using the interpolated normals and tangents, we need to make them orthonormal first.

By the way, we have a well-known solution for it already, i.e. Gram-Schmidt Orthonormalization, which was mentioned by Apatriarca along with source code.




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