Skip to main content

Why is the normal transformation a inverse transpose? (6)


Normal vector as a covariant vector

This third explanation includes how to transform normal vectors. This explanation is based on one of my favorite book[5] by Sugihara. This explanation is a bit formal and less intuitive compare to the first and the second explanation. If you are not interested in a formal explanation, you can skip this section.

First I would like to introduce the affine transformation.

Affine transformation is one of linear transformations. This is quite often used in computer graphics area. Affine transformation transform a line to a line and keep the ratio on a line. If we include a degenerated case, a triangle is always transformed into a triangle. Assume a representation of a three-dimensional affine transformation is a 3x3 matrix. This transformation is a transformation between two coodinate systems. Therefore, I think an object deformation by the transformation is a secondary effect. As a result, we can deform an object. However, this is a transformation of coordinate systems, we can not deform an object arbitrarily. In general, this transformation is a combination of scaling, roration, translation, and shearing. This means, an affine transformation can not transform a triangle to a circle.

If we consider an affine transformation as an coordinate transformation, our interest is how the coordinate of a point P represented in the two different coordinate systems \Sigma and \Sigma'. Note, P itself doesn't move as shown in Figure[3]. P doesn't move, but the coordinates are changed depends on the coordinate system.  These coordinate systems may not have perpendicular basis. They may change the distance depends on the axis direction. For example, x direction is two times magnified to y direction.
Figure 3. A matrix A transforms the coordinate system \Sigma to \Sigma'. Note: the point P doesn't move, but its coordinates representation may differ depends on the coordinate system.

It's a bit cumbersome, but I will write down how the coordinates of P is represented in the coordinate system \Sigma and \Sigma'.

A point P is represented in a coordinate system \Sigma,
and is represented in a coordinate system \Sigma',
A representation means you can define the concrete coordinates, e.g., (1,1,0)^T. On the other hand, if you just have a point P, you don't need to know what is the exact coordinates. Even you don't need to know this point P is in 2-dimensional space or 3-dimensional space. The relationship between a point P and its representation is similar to a linear operator T and its representation matrix M. I can also think the relationship between an interface and its implementation in a programming language context. (I just think the next step of this analogy is related with Futamura projections[1], but it is beyond this article and my understanding is not enought to explain it yet.)
Labels:

Comments

Post a Comment

Popular posts from this blog

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...
Post a Comment
Read more

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...
Post a Comment
Read more

Tezuka Osamu's Black Jack, "Shrinking"

I like several novel authors. My first favorite author is probably Teduka, Osamu. I still love him. The list grows by adding Hoshi, Shinichi, Agatha Christie, Hermann Hesse, and so forth. My first favorite article of Tezuka was Atom as most of the (boy's) Tezuka fans did. But my favorite is Black Jack. I try to summarize one story, it is still quite vivid in my memory. I first read this story when I was 13 - 15 years old. I re-read it at least several times since Black Jack is composed of many short episodes. The title should be "ちぢむ (SHRINKING)" or it might be "縮む(Shrinking)". (It is not so convenient to translate this to English, since English does not have a system to say the exact same word in several ways. So I just simulate it with capital letters.) Black Jack is a genius surgeon, but he does not have the license. In short, his medical activity is illegal. His skill is top level in the world, but, the fee is also out-of-law expensive. In the story ...
Post a Comment
Read more