- #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)

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!

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)

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!