I Visualising rotation in 3-D space

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Visualizing the rotation of the x, y, z axes about a specified axis can be challenging, particularly when trying to understand how the axes change after a rotation, such as 30 degrees. Drawing a diagram with the axis of rotation and considering the fixed length and angle of rotated vectors can aid in comprehension. Online tools like GeoGebra and Desmos offer interactive ways to visualize these transformations, which can enhance understanding. It's noted that vectors parallel to the rotation axis remain unchanged, while those on the disk surface maintain their magnitudes but change direction. Engaging with 3D software or physical models can further solidify the conceptual grasp of these rotations.
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How do you visualise a rotation of the ##x,y,z## axes via an angle ##\theta## about an axis passing through the origin and a point, say ##\text{P}(a,b,c)\quad(a<b<c)##? Let's assume the rotation is anti-clockwise as seen from the origin towards the point ##\text{P}##. Assume ##\theta<\dfrac{\pi}{2}##.

My question is - what would the ##x,y,z## axes look like following the rotation?

I am just not able to do it.
1751299808781.webp
I have very little clue as to how to imagine (visualise) the rotation and how the axes will look.

All I can do is to draw the image of what I mean by the task. Of course, this is before the rotation takes place.

In the diagram, ##\mathbf{OP}## is the axis of rotation.

For simplicity, we may give a value like ##30^{\circ}## to the amount of rotation.

I'd be grateful for any help.
 
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brotherbobby said:
I am just not able to do it.
There are free online tools for this, like GeoGebra.

If you want to draw it by hand, note that every rotated vector has a fixed length and fixed angle to the rotation axis. So each rotated vector stays within a certain cone mantle, centered around the rotation axis. Drawing the base of that cone might help.
 
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brotherbobby said:
TL;DR Summary: How do you visualise a rotation of the ##x,y,z## axes via an angle ##\theta## about an axis passing through the origin and a point, say ##\text{P}(a,b,c)\quad(a<b<c)##? Let's assume the rotation is anti-clockwise as seen from the origin towards the point ##\text{P}##. Assume ##\theta<\dfrac{\pi}{2}##.

My question is - what would the ##x,y,z## axes look like following the rotation?

I am just not able to do it.

View attachment 362717I have very little clue as to how to imagine (visualise) the rotation and how the axes will look.

All I can do is to draw the image of what I mean by the task. Of course, this is before the rotation takes place.

In the diagram, ##\mathbf{OP}## is the axis of rotation.

For simplicity, we may give a value like ##30^{\circ}## to the amount of rotation.

I'd be grateful for any help.
Read about Euler angles here.
 
kuruman said:
Read about Euler angles here.
Euler's angles are useful in that they help you find the answer. But you don't get a feel of "how" in real time.
I'd like a way to visualise directly; what happens to a point (or several points on a line) as one effects a rotation about a given axis.
 
A.T. said:
There are free online tools for this, like GeoGebra.

If you want to draw it by hand, note that every rotated vector has a fixed length and fixed angle to the rotation axis. So each rotated vector stays within a certain cone mantle, centered around the rotation axis. Drawing the base of that cone might help.
I could visualise what you mean. So thank you for that. Indeed, the ##x## axis starts with a precise angle to the axis of rotation and, if we take any point on the ##x## axis, the line would continue having a fixed length throughout the rotation. The angle would remain the same too.
In my problem above, clearly the rotation is not taking along any of the coordinate planes. I cannot visualise how the ##x## axis would look after a rotation of ##30^{\circ}## about the axis given. 😌
 
brotherbobby said:
But you don't get a feel of "how" in real time.
I think playing around in 3D software will give you an intuition for that.
 
brotherbobby said:
I'd like a way to visualise directly; what happens to a point (or several points on a line) as one effects a rotation about a given axis.
I used to hold a child's toy globe in my hands, now I imagine an Earth globe and rotate it in my head and hands.

Null Island in the Atlantic Ocean is +x, the Indian Ocean contains +y, while the North Pole is +z.
https://en.wikipedia.org/wiki/Null_Island
 
robphy said:
It might help to draw a disk centered at the origin whose normal is along the axis of rotation.

www.desmos.com/3d/gflyvyyuoh

That looks extremely good. I am still struggling with the problem and, despite your efforts, I might continue to do so.
Still, thank you very much. Not only do I want to understand how vectors (and co-ordinate axes) rotate under arbitrary transformations about an axis, I am also eager to learn the 3-D drawing on ##\verb|Desmos.com|##
 
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robphy said:
It might help to draw a disk centered at the origin whose normal is along the axis of rotation.

www.desmos.com/3d/gflyvyyuoh
If your axis of rotation is along the z-axis, the disk represents the unit-disk on the xy-plane.
Vectors parallel to the axis of rotation don't change.
Vectors parallel to the disk surface keep their magnitudes but change their directions.
A general vector can be decomposed into the sum of a vector-parallel and a vector-perpendicular to the axis of rotation. (Thus, for a general vector, we draw an associated right-triangle.)

Now, as you change your axis of rotation (away from the z-axis),
you can see the change of the associated disk (off the xy-plane).

Upon rotation about that axis, a general vector decomposed with the associated right-triangle
will show the height of the triangle (parallel to the axis of rotation) unchanged and
the projection (the shadow) onto the disk have a constant size, but different direction according to the rotation.



I learned tools like desmos by modifying examples.
Try turning off various features by long-clicking the colored-circles.
Try modifying the expressions. (You can always undo or reload. You can also save your own copy.)
 
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