Coordinate Transformations

In summary, the conversation discusses finding transformation matrices for rotating coordinate systems in classical mechanics, specifically in linear algebra. The person is having trouble visualizing the rotations, and asks for help in understanding the concept. They provide two examples and matrices they have come up with, and someone responds with images that can help with visualization.
  • #1
don_anon25
36
0
These problems are actually for my classical mechanics class, but they are linear-algebra based. I can construct a transformation matrix, but I have trouble visualizing the rotations, particularly in 3-space. So if someone could help me get a pictorial idea of what's actually happening, then the problems would be much easier!

1) Find the transformation matrix that rotates the x3 (z) axis of a regular coordinate system 45 degrees toward x1 (x) around the x2 (y) -axis.
Here's the matrix I got for this one:
1 0 0
0 1 0
sqrt2/2 -sqrt2/2 sqrt2/2


2) Find the transformation matrix that rotates a rectangular coordinate system through an angle of 120 degrees about an axis making equal angles with the original three coordinate axes.
Here's the matrix I came up with for this one:
-.5 sqrt3/2 .5
.5 -.5 sqrt3/2
sqrt3/2 .5 -.5

Thanks!
 
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  • #2
A couple images are posted here:

http://www.akiti.ca/RotateTrans.html

They are not exactly the transformation you have described but they should help you picture what is going on. And you should be able to easily modify the axes for your own application (I don't think your transformation matrix is correct).

Regards,


Duncan
 
  • #3


Hi there,

Sure, I'd be happy to help you visualize these rotations. In both cases, we are rotating the coordinate system in three-dimensional space. So to visualize this, let's start by drawing a 3D Cartesian coordinate system with the x, y, and z axes labeled. This will serve as our original coordinate system.

1) For the first problem, we are rotating the z-axis (x3) 45 degrees towards the x-axis (x1) around the y-axis (x2). To visualize this, imagine a point on the z-axis, say (0,0,1), which represents a unit vector in the z-direction. Now, as we rotate the coordinate system, this point will also rotate. The new position of this point after the rotation will be (sqrt(2)/2, 0, sqrt(2)/2), as you correctly calculated in your matrix. This corresponds to a vector that is 45 degrees from both the z and x axes. Similarly, any other point on the z-axis will also rotate 45 degrees towards the x-axis. This will result in a new coordinate system with the z-axis at an angle of 45 degrees from the original z-axis.

2) For the second problem, we are rotating the coordinate system through an angle of 120 degrees about an axis that makes equal angles with the original three axes. This means that the new axis will be at an angle of 60 degrees with each of the original axes. So to visualize this, imagine a point on the x-axis, say (1,0,0), which represents a unit vector in the x-direction. After the rotation, this point will now be at (0.5, sqrt(3)/2, 0.5), which corresponds to a vector that is 120 degrees from the original x-axis. Similarly, points on the y and z axes will also rotate 120 degrees. This will result in a new coordinate system where all three axes are rotated by 120 degrees from the original axes.

I hope this helps you visualize the rotations and make the problems easier for you. Good luck with your classical mechanics class!
 

1. What is a coordinate transformation?

A coordinate transformation is a mathematical process used to convert coordinates from one coordinate system to another. This is often necessary when working with data or equations that are represented in different coordinate systems, such as Cartesian coordinates, polar coordinates, or geographic coordinates.

2. Why are coordinate transformations important in science?

Coordinate transformations are important in science because they allow us to understand and analyze data in different coordinate systems, providing different perspectives and insights into the same information. They also allow us to make connections and comparisons between different datasets that may be represented in different coordinate systems.

3. What are some common types of coordinate transformations?

Some common types of coordinate transformations include translations, rotations, and scaling. These transformations are used to shift, rotate, or resize coordinates in a given coordinate system to match the coordinates in another system.

4. How are coordinate transformations used in real-world applications?

Coordinate transformations are used in a variety of real-world applications, including mapping and navigation systems, satellite imagery, and geographic information systems. They are also used in physics and engineering to analyze and solve problems involving different coordinate systems.

5. What are some challenges with coordinate transformations?

One of the main challenges with coordinate transformations is the potential for error. Small errors in the transformation process can lead to significant discrepancies in the final results. Additionally, some coordinate systems may have more complex transformations, making them more difficult to work with and potentially introducing more errors.

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