Why is the normal transformation a inverse transpose? (2)

What is the problem? What is the transformation matrix of normal?

We use transformation matrices every day when we move objects in a computer. Current the state of the art DCC (digital contents creation) software usually represents objects with triangles or polygons. Each vertex of the triangles or polygons usually has its coordinates. When we rotate or move each vertex, we apply a transformation matrix on each vertex. A vertex is usually three dimensional vector in computer graphics.

We can define a normal vector for each triangle. A normal vector points out to which direction a triangle face is oriented. This normal vector is also a three dimensional vector. In a 3D computer graphics system, normal vectors are important since we need these normal vectors to compute how bright the surfaces are. Because a usual vector can be transformed by a matrix, it seems straightforward to use the same matrix to transform a normal vector. However, this fails. But why? The article is all about this ``why?''

Why an usual transformation matrix fails on a normal vector?

According to [3], an explanation by Eric Haines is quite good. The book [6] has the same explanation, I see that is a great explanation. A similar explanation can also found in [2]. Figure 1 shows the similar explanation.

Figure1. Scaling on a normal break the normal.

Figure 1 shows a three dimensional plane standing straight (standing z up direction) and the view point is from the top (view is the z minus direction). Simply, We are looking down a wall from the top. This wall has a (1,1,0) normal vector. Let's think to apply the following transformation matrix M. This matrix M magnifies x direction twice than other directions.

Now you see the wall is double sized in x direction, but, if we apply this matrix to the normal, the normal is no longer normal vector of this wall. In the left figure of Figure 1, the normal vector is perpendicular to the wall, but, in the right figure, the transformed normal vector is no perpendicular to the wall any more. This is the problem.

Why cannot we transform the normal vector same as usual vectors?

Comments

Why A^{T}A is invertible? (2) Linear Algebra

Why A^{T}A has the inverse Let me explain why A^{T}A has the inverse, if the columns of A are independent. First, if a matrix is n by n, and all the columns are independent, then this is a square full rank matrix. Therefore, there is the inverse. So, the problem is when A is a m by n, rectangle matrix. Strang's explanation is based on null space. Null space and column space are the fundamental of the linear algebra. This explanation is simple and clear. However, when I was a University student, I did not recall the explanation of the null space in my linear algebra class. Maybe I was careless. I regret that... Explanation based on null space This explanation is based on Strang's book. Column space and null space are the main characters. Let's start with this explanation. Assume x where x is in the null space of A . The matrices ( A^{T} A ) and A share the null space as the following: This means, if x is in the null space of A , x is also in the n...

Gauss's quote for positive, negative, and imaginary number

Recently I watched the following great videos about imaginary numbers by Welch Labs. https://youtu.be/T647CGsuOVU?list=PLiaHhY2iBX9g6KIvZ_703G3KJXapKkNaF I like this article about naming of math by Kalid Azad. https://betterexplained.com/articles/learning-tip-idea-name/ Both articles mentioned about Gauss, who suggested to use other names of positive, negative, and imaginary numbers. Gauss wrote these names are wrong and that is one of the reason people didn't get why negative times negative is positive, or, pure positive imaginary times pure positive imaginary is negative real number. I made a few videos about explaining why -1 * -1 = +1, too. Explanation: why -1 * -1 = +1 by pattern https://youtu.be/uD7JRdAzKP8 Explanation: why -1 * -1 = +1 by climbing a mountain https://youtu.be/uD7JRdAzKP8 But actually Gauss's insight is much powerful. The original is in the Gauß, Werke, Bd. 2, S. 178 . Hätte man +1, -1, √-1) nicht positiv, negative, imaginäre (oder gar um...

Why parallelogram area is |ad-bc|?

Here is my question. The area of parallelogram is the difference of these two rectangles (red rectangle - blue rectangle). This is not intuitive for me. If you also think it is not so intuitive, you might interested in my slides. I try to explain this for hight school students. Slides: A bit intuitive (for me) explanation of area of parallelogram (to my site, external link) .

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