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Why is the normal transformation a inverse transpose? (1)


Abstract

Several books[2,6] explained the normal vector transformation matrix is $(M^{-1})^{T}$. I always forget this formula. This time I understand it a bit in three different ways, so I will write them down here.

Introduction

Assume matrix M is applied to a vector where the M is a coordinate transformation matrix. For example, M could be a translation, rotation, scaling, and so forth. To transform a position vector, we can just multiply this matrix M. However, we may fail when we transform a normal vector by just multiplying the matrix M. Several books mentioned normal transformation matrix should be $(M^{-1})^{T}$[2,6].

In this article, I would like to mention about the following three issues:

  • What is the problem? What is the transformation matrix of normal?
  • Why may multiplying matrix M fail?
  • Why is it an inverse of transpose of matrix M?
(The references will be shown up at the end of this series.)

Next time, I would like to talk about the problem.

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