Rotation of a point in R3 about the y-axis

In summary, the conversation discusses a problem with visualizing a rotation around the ##y##-axis in a right-handed cartesian coordinate system. The equations for the change in the point's ##x## and ##z## coordinates are provided, but the speaker struggles to understand the visual representation for the equation of ##x##. Through discussion and visual aids, the speaker is able to find a solution and shares it with others. They also mention a more general equation, Rodrigues' rotation formula, that can be applied in similar cases. Overall, the conversation highlights the importance of understanding right-handed and left-handed coordinate systems and using visual aids to better understand mathematical concepts.
  • #1
NatFex
26
3
Hello,

I'm having a visualisation problem. I have a point in R3 that I want to rotate about the ##y##-axis anticlockwise (assuming a right-handed cartesian coordinate system.) I know that the change to the point's ##x## and ##z## coordinates can be described as follows:

$$z = z'\cos\theta-x'\sin\theta$$
$$x = x'\cos\theta+z'\sin\theta$$

My problem is that I only seem to be able to know this "as fact" and I am trying to draw a visual aid to help me see how this has come about. I produced the following diagram: (I know the rotation looks clockwise but it's anticlockwise as the ##y##-axis is coming out of the page/screen)

34y1oa0.png

As you can see, the first equation for the ##z## coordinate of the point is derivable by looking at the heights (blue edges) of the pale yellow triangles. The longer blue edge is ##z'\cos\theta## by simple trig and the shorter one is ##x'\sin\theta##. Take the difference of the 2 to work out the ##z## coordinate (where the purple line touches the ##z##-axis) et voila.

The issue is I am struggling to come up with a similar visualisation for ##x##. I just can't find the right triangles to make that sum. I've burned through ~10 sheets of paper already. It's definitely one of those "need-to-disengage-to-see-clearly-again" situations because I have definitely done this before, but I figured someone on here might speed this tedious process up and guide me on how to visually represent the equation for ##x##. Either verbally point me in the right or scribble on top of my drawing, I'm not bothered. Thanks!
 

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  • #2
What makes you say the y-axis is coming out of the screen ? Do you know about right-handed and left-handed coordinate systems ?
 
  • #3
Another approach to understanding this could be to understand a more general exposition like Rodrigues' rotation formula [1] and then just apply the values that apply for your case. At least, for practical applications this a very handy equation.

[1] https://en.wikipedia.org/wiki/Rodrigues'_rotation_formula
 
  • #4
Visual aid: ignore y. Set up graph with x and z as axes. Then put in a pair of perpendicular lines through the origin and label them x' and z'. Label the angle between x and x' as ##\theta## and compute the relationships
 
  • #5
BvU said:
What makes you say the y-axis is coming out of the screen ? Do you know about right-handed and left-handed coordinate systems ?
I was confused by your comment at first but you're right, I have it the wrong way around. OK, so the ##y##-axis (as per right-handed cartesian system) is going into the screen. The only way to make my diagram still relevant is to instead say that we're rotating clockwise around ##y##, so my little comment about that is completely irrelevant. Will edit the OP as necessary. EDIT: Seems I can't actually edit it any more.

mathman said:
Visual aid: ignore y. Set up graph with x and z as axes. Then put in a pair of perpendicular lines through the origin and label them x' and z'. Label the angle between x and x' as ##\theta## and compute the relationships
If you open the spoiler you will see that I had an attempt at exactly what you describe. Managed to get one (but not both) of the equations that way. This thread was made because I got stuck
 
  • #6
Hi all, excuse the double post but since I just managed to find what I was looking for I thought I'd share in case anybody else finds seeing where the equations in the OP come from

Quick scribble since I can't be bothered to make it as neat as my original diagram at the moment:

a0uux5.png

The ##x##-coordinate is obtained by adding the 2 widths (neon green lines) of the fuchsia triangles.
 

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  • #7
NatFex said:
Hi all, excuse the double post but since I just managed to find what I was looking for I thought I'd share in case anybody else finds seeing where the equations in the OP come from

A couple of observations. You have taken a point in the ##x'-z'## system and calculated its coordinates in the ##x-z## system. Effectively, you have rotated your axes by ##\theta##. The primed axes are the unprimed axes rotated clockwise by ##\theta##.

This gives the same form of the equations as taking a vector in the ##x-z## system, with coordinates ##(x, z)##, and rotating it anticlockwise by ##\theta## and giving the coordinates of the new vector as ##(x', z')##.

Note that the simplest way to generate the coordinates is to use the sum of angles formulae for sine and cosine. If a vector has a polar angle of ##\alpha## anticlockwise from the x-axis and is rotated by an angle ##\theta## anticlockwise, then its new polar angle is ##\alpha + \theta##. If you expand the expressions for ##\cos(\alpha + \theta)## and ##\sin(\alpha + \theta)##, you get the required formula for the coordinates of a rotated vector.
 

1. What is the y-axis in R3?

The y-axis in R3 is the vertical line that extends infinitely in both positive and negative directions. It is one of three axes that form a three-dimensional coordinate system, along with the x-axis (horizontal) and z-axis (depth).

2. How is a point rotated about the y-axis in R3?

A point in R3 can be rotated about the y-axis by keeping the y-coordinate constant and changing the x and z coordinates using the following formula:

x' = x*cosθ - z*sinθ

y' = y

z' = x*sinθ + z*cosθ

Where θ is the angle of rotation.

3. What is the purpose of rotating a point about the y-axis in R3?

Rotating a point about the y-axis in R3 can be used to change the orientation or position of an object in three-dimensional space. It is commonly used in computer graphics and animation to create realistic movements and transformations.

4. How does the direction of rotation affect the result?

The direction of rotation affects the result of rotating a point about the y-axis in R3. If the rotation is clockwise, the x and z coordinates will decrease. If the rotation is counterclockwise, the x and z coordinates will increase. This can be visualized by imagining standing on the positive y-axis and looking down towards the origin.

5. Can a point be rotated about the y-axis in R3 more than once?

Yes, a point in R3 can be rotated about the y-axis multiple times. Each successive rotation will result in a new position for the point. It is important to note that the angle of rotation will also change with each successive rotation, as the point's new position will be relative to its previous position.

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