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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 righthanded cartesian coordinate system.) I know that the change to the point's ##x## and ##z## coordinates can be described as follows:
$$z = z'\cos\thetax'\sin\theta$$
$$x = x'\cos\theta+z'\sin\theta$$
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 "needtodisengagetoseeclearlyagain" 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 righthanded cartesian coordinate system.) I know that the change to the point's ##x## and ##z## coordinates can be described as follows:
$$z = z'\cos\thetax'\sin\theta$$
$$x = x'\cos\theta+z'\sin\theta$$
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 "needtodisengagetoseeclearlyagain" 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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