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I have bit of mechanical problem in real life that I need to solve by math. Not sure is this trigonometric problem or something else but here's what I have.

enter image description here I'm not mathematician so don't blame me if I used incorrect markings on illustration above or not using proper terminology. Feel free to edit title or whatever is needed because this is the best I could do to explain. Anyway, I will try to explain the problem in my own way.

I have 3 lines, R1A, AB and R2B in 2D space (XY). Line R1A is rotating around point R1 by angle α in negative direction (CCW). Line R2B is rotating around point R2 by angle β in positive direction (CW). Points A and B are connected with a line of known length (c - distance AB has constant value). Coordinates of points in 2D space A, R1 and R2 are known and angles of rotation α and β are known.

How to find coordinates of point B (distance R2B works too instead) so when we rotate line R1A by α angle, line R2B rotates by β angle?

To visualize this (white values are known, orange is not):

enter image description here

Wh1T3h4Ck5
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  • If $R1$ and $R2$ are fixed in place, then just set the arc length travelled by $A$ equal to the arc length travelled by $B$. Solve for angle $\beta$. – Doug Jan 16 '24 at 18:37
  • @Doug, Distance AB must be constant, like a rod connecting two mechanical parts. I don't know where point B is and not sure how to do that. – Wh1T3h4Ck5 Jan 16 '24 at 18:44
  • Do you know the radii of the circles? – Vasili Jan 16 '24 at 21:27
  • @Vasili, Only of left one... can be calculated as a distance between known coordinates of R1 and A, but coordinates of point B are unknown, so radius of another circle around R2 is also unknown. My goal is to find formula to get coordinates of point B. Course, max angle R1AB and ABR2 is 180 degrees, so α and β are bit limited. In real case α and β can't go over 30 degrees for given direction of rotations. – Wh1T3h4Ck5 Jan 16 '24 at 22:09
  • So if the left circle has radius $r_1$ and the right one has radius $r_2$, the point $A$ travels $\alpha \cdot r_1= \beta \cdot r_2$ which should give you $r_2$ – Vasili Jan 16 '24 at 23:56
  • @Vasili, Makes sense. Let me try it and do some tests to check does it work. – Wh1T3h4Ck5 Jan 17 '24 at 00:00
  • @Vasili Yep, that worked and guess that's what Doug was talking about in the first comment. Thanx a million guys, saved my day! Consider posting that as answer so I can accept that as solution. – Wh1T3h4Ck5 Jan 17 '24 at 15:38
  • @Wh1T3h4Ck5: Glad it worked! I converted my comment into an answer. – Vasili Jan 17 '24 at 19:42

1 Answers1

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As @Doug also noted, we have a rigid rod $c$ with endpoints $A$ and $B$. Therefore, if you displace point $A$, point $B$ will be displaced by the same distance (assuming no other forces are applied/relevant). Thus if the left circle has radius $r_1$ and the right one has radius $r_2$ and we use radian measure of angles, the following equation can be used to find $r_2$: $$\alpha \cdot r_1=\beta \cdot r_2$$

Vasili
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