Given a normalized 3D vector, here’s an efficient method for computing a full basis. The computed basis is axis aligned if the input vector is axis aligned.

void ComputeBasis(const Vec& a, Vec* b, Vec* c)

{

// Suppose vector a has all equal components and is a unit vector: a = (s, s, s)

// Then 3*s*s = 1, s = sqrt(1/3) = 0.57735. This means that at least one component of a

// unit vector must be greater or equal to 0.57735.

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if (Abs(a.x) >= 0.57735f)

b->Set(a.y, -a.x, 0.0f);

else

b->Set(0.0f, a.z, -a.y);

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b = Normalize(b);

*c = Cross(a, *b);

}

In SSE land you can eliminate the branch using a select operation.

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