2012-05-20

Why is the normal transformation a inverse transpose? (7)


Let coordinate \Sigma 's the origin O, coordinate \Sigma' 's origin O', then we can think about the coordinates of O'O in the coordinate system \Sigma' that is represented as:
I would like to have a comment of this why this matters. In some graphics system, e.g., OpenGL, we can move or distort objects by applying transformation matrix.  This looks like the object is moved or is distorted. This interpretation is possible, but, here we did not move the objects, but we changed the coordinate system. This is rather how we interpret the result. I prefer this interpretation since we can think changing coordinate system without objects. You can still think the applying transformation matrix is an applying an operator, but, object is just one subject to operate. I would like to think rather about operator itself. Then we can concentrate operator itself, the transformation matrix itself. In Figure 3, there is a point P, this point actually doesn't move in the space. But the coordinates was changed. Let me have an example as a city map. We can make any point or landmark as the origin of the city map. For instance, we can see the Zoologischer Garten as the origin of the city Berlin (in Germany) map. Also we can have an map that origin is Alexanderplatz station. The coordinates of other landmarks are not the same between these two maps.  But Zoologischer Garten never moved. It is just in the different coordinate system. We can define a coordinate system arbitrarily. Therefore, which coordinate system we can use is our choice. The important thing here is we can convert one coordinate system to another coordinate system. Then, there is no problem to choose your favorite coordinate system. This is the motivation and reason that I am explaining transforming coordinate systems. (Linear) Transforming coordinate system is generally done by a transformation matrix.

Transforming coordinate system follows the next equation:
If A is regular, there exists an inverse matrix of A,
Note,
I think this is clear when you see Figure 3. The origin movement is just a translation, the inverse is movement of the opposite direction. Therefore, we can rewrite the equation to:

On the other hand, when a normal vector n is defined in a plane which has a point P, it is defined in the coordinate system \Sigma as:

When this normal is transformed in the coordinate system \Sigma', it is defined as:
Let plug in Equation (1) into (2).
Let's compare the Equation (3) and Equation (4):
Equation (5) shows how the normal vector is transformed. The vectors that are transformed like this are called ``covariant vector.'' Usual vectors are called ``contravariant vector.'' co (together) variant (changing) vector changes following the coordinate system's transformation. On the other hand, usual vectors changed their representation, but not changed as the vector itself. It is like the point P changed the representation of the coordinates, but the point itself is actually never moved. Therefore, these vectors are called contra (against) variant (change) vector.

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