## Conclusion

I think it is not exactly correct that the normal transform is an inverse transpose. A normal vector is not a column vector, the transform is a left inverse as Equation 5 shows. The transpose comes from the implementation of the vector class. This should be a left inverse as a rigorous formulation. I don't say the references [2,3,6] are incorrect, but I prefer the references [4,5] since they distinguish these two kind of vectors.

However, the references [2,3,6] have an easier to understand explanations. Especially, [2,3,6] shows why normal transformation doesn't work with applying A intuitively. I like these explanations more. Here I would like to highlight the difference of a usual vector and a normal vector, they are a column vector or a row vector. By the way, contravariant vector means doesn't-change-vector, but, its representation can change. It seems I should explain the difference of these vectors are related with inner product. I would like to study more on this and hopefully I could explain this better one day.

I find it is interesting theme that what is not change by the coordinate transformation. For example, Euclidean distance doesn't change even we translate the coordinate system. That is related inner product. Another example is magnetic field of Maxwell equation. Einstein got a hint of relatively theory because of any coordinate transformation should not change the divergence (no magnetic monopole). I hope I can write something about this in the future.

## References

[1] Yoshihiko Futamura, http://en.wikipedia.org/wiki/Partial_evaluation

[2] Matt Pharr, Greg Humphreys, ``Physically Based Rendering, Second Edition: From Theory To Implementation'', Morgan Kaufmann, 2010

[3] Philip Schneider, David H. Eberly, ``Geometric Tools for Computer Graphics'', Morgan Kaufmann, 2002

[4] Gilbert Strang, ``Introduction to Linear Algebra, 4th Edition'', Wellesley-Cambridge Press, 2009

[5] Koukichi Sugihara, ``Guraphics no suuri (Mathematical theory of graphics)'', Kyouritu shuppan, 1995 (杉原厚吉, グラフィックスの数理, 共立出版, 1995)

[6] Tomas Akenine-Moller, Eric Haines, Naty Hoffman, ``Real-Time Rendering (2nd Edition)'',
A K Peters/CRC Press, 2002