Suppose I have a pantograph, as illustrated below.

The base is fixed in that it doesn't move around in space, but arms a and b are able to rotate freely around it, thus moving the tip around according to the angle of a and b from the vertical.

Is there an equation that will describe the x/y position of the tip, given the angles of the arms a and b relative to the vertical, and vice-versa?

If it makes it easier to use angles relative to the horizontal, or something else, that's also fine.

And if this isn't actually an example of a pantograph but is something else, please feel free to edit the question.



Assuming that this contraption will always assume the shape of a rhombus, we have that the line connecting the tip to the fixed base will always bisect the angle between the two arms, and also bisect the line segment connecting the tips of the two arms. You can use this to draw a right triangle from the middle of said line segment to the fixed base to the end of either arm.

Thus, if the arms have length $l$ and their angles are $\theta_a$ and $\theta_b$ from the horizontal (with the assumption that $\theta_a\gt \theta_b$), we may calculate using the mentioned right triangle that the distance from the fixed base to the tip is equal to $$2l\cos\frac{\theta_a-\theta_b}{2}$$ Now, since the angle formed between the horizontal and the tip is given by $\frac{\theta_a+\theta_b}{2}$, if we let the fixed base be at the origin, we have the following x- and y-positions for the tip: $$x=2l\cos\frac{\theta_a-\theta_b}{2}\cos\frac{\theta_a+\theta_b}{2}$$ $$y=2l\cos\frac{\theta_a-\theta_b}{2}\sin\frac{\theta_a+\theta_b}{2}$$

  • $\begingroup$ Thanks, I think that's what I need (though I realised I also need to go the other way, from x/y to the angles, but I might be able to rearrange the equations to do that). $\endgroup$ Dec 20 '18 at 14:51
  • 1
    $\begingroup$ @DanieleProcida No problem! If this answer is helpful, don't forget to $\uparrow$ and $\checkmark$! $\endgroup$ Dec 20 '18 at 15:02

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