I Visualising rotation in 3-D space

[brotherbobby]

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.

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.

 

[A.T.]

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.

 

[kuruman]

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.

 

[brotherbobby]

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.

 

[brotherbobby]

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.

 

[A.T.]

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.

 

[Baluncore]

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]

It might help to draw a disk centered at the origin whose normal is along the axis of rotation.

www.desmos.com/3d/gflyvyyuoh

 

[brotherbobby]

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 $\verb|Desmos.com|$

 

[robphy]

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.

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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.)