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Normal for bump mapping

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I was looking at a nice tutorial on how to perform simple 2D bump mapping.

There is just one little thing I don't get:

If we know X and Y of normal, we can calculate it's Z
(as normals are unit vectors. So Z is simply 1-sqrt(X^2 + Y^2) ).

If I do the maths, as the normal is a vector unit then:

1 = sqrt(x^2 + y^2 + z^2)
1 = x^2 + y^2 + z^2
1 - x^2 - y^2 = z^2
z = sqrt(1 - (x^2 + y^2))

Which is not the same formula as in the tutorial...
Could anyone tell me where the problem is?


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