# How can I calculate the position of pantograph's tip from the angles at its base?

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.

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: $$x=2l\cos\frac{\theta_a-\theta_b}{2}\cos\frac{\theta_a+\theta_b}{2}$$ $$y=2l\cos\frac{\theta_a-\theta_b}{2}\sin\frac{\theta_a+\theta_b}{2}$$
• @DanieleProcida No problem! If this answer is helpful, don't forget to $\uparrow$ and $\checkmark$! Dec 20 '18 at 15:02