Coordinate transformation - Rotation

In summary, the author derives the old basis unit vectors in terms of new basis vectors by projecting the old basis onto the new. This can be done through trigonometry, where the old basis vector forms the hypotenuse of a right triangle and its projections onto the new coordinate directions can be found using the definitions of sine and cosine. Alternatively, the author defines a rotation as the transformation between two orthonormal bases and uses the orthonormality conditions to derive a set of relations among the coefficients, leaving one free parameter (the angle of rotation). The interpretation of this angle can be made by looking at the inner product between the old and new basis vectors.
  • #1
Quanta
ort1.jpg

ort2.jpg

How author derives these old basis unit vectors in terms of new basis vectors ? Please don't explain in two words.

[itex] \hat{e}_x = cos(\varphi)\hat{e}'_x - sin(\varphi)\hat{e}'_y[/itex]
[itex] \hat{e}_y = sin(\varphi)\hat{e}'_x + cos(\varphi)\hat{e}'_y[/itex]​
 
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  • #2
How about one word? Trigonometry.

It is just a matter of projecting the old basis on the new. Since the new basis is orthonormal, it holds that ##\vec e_i = (\vec e_i\cdot\vec e’_j)\vec e’_j##.
 
  • #3
Orodruin said:
How about one word? Trigonometry.

It is just a matter of projecting the old basis on the new. Since the new basis is orthonormal, it holds that ##\vec e_i = (\vec e_i\cdot\vec e’_j)\vec e’_j##.

We obtain:

[itex] \vec{e}_i = (\vec{e}_i \cdot \vec{e}'_j)\vec{e}'_j [/itex]
[itex] \vec{e}_i = | \vec{e}_i | | \vec{e}'_j |cos(\frac{\pi}{2}+\varphi) \vec{e}'_j [/itex]
[itex] \vec{e}_i = -sin(\varphi)\vec{e}'_j [/itex]

Right ? But we didn't get transformation formula.
 
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  • #4
No, you have not done it correctly. You cannot have different free indices on the different sides of the equation. Also note that repeated indices must be summed over.
 
  • #5
Orodruin said:
No, you have not done it correctly. You cannot have different free indices on the different sides of the equation. Also note that repeated indices must be summed over.

[itex] \vec{e}_1 = | \vec{e}_1 | | \vec{e}'_1 |cos(\varphi) \vec{e}'_1 + | \vec{e}_1 | | \vec{e}'_2 |cos(\frac{\pi}{2}+\varphi) \vec{e}'_2[/itex]
[itex] \vec{e}_1 = cos(\varphi) \vec{e}'_1 -sin(\varphi)\vec{e}'_2 [/itex]

[itex] \vec{e}_2 = | \vec{e}_2 | | \vec{e}'_1 |cos(\frac{\pi}{2}-\varphi) \vec{e}'_1 + | \vec{e}_2 | | \vec{e}'_2 |cos(\varphi) \vec{e}'_2[/itex]
[itex] \vec{e}_2 = sin(\varphi) \vec{e}'_1 +cos(\varphi)\vec{e}'_2 [/itex]

Now am I right ?

But I want to understand in projection language (trigonometry) how author derives these formulas, but I didn't find explanation of unit vector basis transformation using trigonometry. I tried but didn't figured out how. Point me right direction.
 
  • #6
The old basis vector is a unit vector that forms the hypothenuse of a right triangle who's other sides are its projections onto the new coordinate directions. The rest is just applying the definitions of sine and cosine in terms of the sides of a right triangle.
 
  • #7
The other way around is to define a rotation as the transformation that relates two orthonormal bases. Looking at the two-dimensional case, you would have
$$
\vec e_x = a \vec e'_x + b \vec e'_y \quad \mbox{and} \quad \vec e_y = c \vec e'_x + d \vec e'_y.
$$
From here you can apply the orthonormality conditions of both systems in asserting that ##\vec e_x^2 = \vec e'^2_x = \vec e_y^2 = \vec e'^2_y = 1## and ##\vec e_x \cdot \vec e_y = \vec e'_x \cdot \vec e'_y = 0##. This will give you a set of relations among the coefficients ##a##, ##b##, ##c##, and ##d## (please derive these relations yourself). You should get three independent relations and since you have four parameters, you will be left with one free parameter (the angle of rotation). You will be able to make the interpretation of the angle of rotation by looking at the inner product between ##\vec e_x## and ##\vec e'_x##.
 
  • #8
Orodruin said:
The old basis vector is a unit vector that forms the hypothenuse of a right triangle who's other sides are its projections onto the new coordinate directions. The rest is just applying the definitions of sine and cosine in terms of the sides of a right triangle.

What you said is evident from the picture, but how ? I drew it many times but cannot get sum of projections that is a old basis unit vector.
 
  • #9
Quanta said:
What you said is evident from the picture, but how ? I drew it many times but cannot get sum of projections that is a old basis unit vector.
I don't understand what you are trying to say here. It is evident from the picture but it is not evident?
 
  • #10
Orodruin said:
I don't understand what you are trying to say here. It is evident from the picture but it is not evident?

How to project [itex]\hat{e}_x[/itex] on [itex]\hat{e}'_x[/itex] and [itex]\hat{e}'_y[/itex] ?

basis.jpg
 
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  • #11
Quanta said:
How to project [itex]\hat{e}_x[/itex] on [itex]\hat{e}'_x[/itex] and [itex]\hat{e}'_y[/itex] ?
Rotate your figure until [itex]\hat{e}'_x[/itex] is horizontal. Then the projections will be obvious.
 

1. What is coordinate transformation?

Coordinate transformation is the process of changing the coordinates of points in a coordinate system to another coordinate system. This is often necessary when working with different coordinate systems, such as Cartesian and polar coordinates.

2. What is rotation in coordinate transformation?

Rotation in coordinate transformation refers to the act of rotating a coordinate system around a fixed point. This is done by changing the orientation of the axes in the coordinate system, while keeping the origin fixed.

3. What is the purpose of coordinate transformation?

The purpose of coordinate transformation is to make it easier to work with different coordinate systems and to solve problems that involve multiple coordinate systems. It allows for the conversion of coordinates between systems, making it possible to transfer information and data between different systems.

4. How is rotation represented in coordinate transformation?

Rotation is typically represented using matrices in coordinate transformation. These matrices describe the transformation of points from one coordinate system to another, taking into account the rotation angle and the axis of rotation.

5. What are some applications of coordinate transformation - rotation?

Coordinate transformation - rotation is used in a variety of fields, including mathematics, physics, engineering, and computer graphics. It is commonly used in navigation and mapping systems, robotics, and 3D modeling. It is also essential in solving problems involving rigid body motion and transformations between different reference frames.

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