I Rotation of a point in R3 about the y-axis

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!
 

Attachments

BvU

Science Advisor
Homework Helper
12,338
2,743
What makes you say the y axis is coming out of the screen ? Do you know about right-handed and left-handed coordinate systems ?
 

Filip Larsen

Gold Member
1,226
160
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
 

mathman

Science Advisor
7,689
389
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
 
26
3
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.

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
 
26
3
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.
 

Attachments

PeroK

Science Advisor
Homework Helper
Insights Author
Gold Member
2018 Award
9,930
3,728
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.
 

Want to reply to this thread?

"Rotation of a point in R3 about the y-axis" You must log in or register to reply here.

Related Threads for: Rotation of a point in R3 about the y-axis

Replies
5
Views
2K
Replies
6
Views
2K
Replies
4
Views
2K
Replies
6
Views
10K
Replies
1
Views
1K
Replies
3
Views
3K
Replies
3
Views
757
Replies
6
Views
3K

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top