Find Arc length from a given R were it intersects 2 given R that have a CP of X0 Y0 and the arc length R X ,Y is known

Question

I have a program for drawing cams that needs the displacement input in Arc Length but the designs are given to me with the displacement’s in a cord length Is there a formula to convert the liner cord length to the arc length with the given dimensions.

Answer 1

First let's solve the problem of finding the length of $arc(AD)$:

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$B$ and $C$ are the centers of the circles. Considering your picture you have all lengths. We have first to find $\angle ABC=arc(AD)/2$ (that must to be in radians) which is given by cossine rule:

$$r^2=R^2+d^2-2Rd\cos(\angle ABC) \Rightarrow \angle ABC = \arccos\left(\frac{R^2+d^2-r^2}{2Rd}\right) \quad (1)$$

After that you can find what you want:

The length that you want is given by:

$$arc(AB)=\frac{arc(AD)-arc(BC)}{2}$$

and both $arc(AD)$ and $arc(BC)$ are given by the equation $(1)$.

I hope that helps.