I need to find the length between an arc and the chord at any given interval along the chord's length. I know the distance at the center of the arc and chord. I also know the chord length and radius. How can I figure out the length of a segment from the chord to the arc (given the ray is 90* to the chord) say 2" from the end of the chord? 1"? 5"?
How do I calculate to get 0.773 as shown in this cad drawing?
Let's call the center of the circle $(0,0)$, with $y$ axis up, and $x$ axis to the right. Then the midpoint of the chord is $(0,39-3)=(0,36)$. The left intersection is then $(-15,36)$. Any point $(x,y)$ on the arc then obeys: $$x^2+y^2=R^2=39^2$$ We can write this point in terms of the left intersection: $$(x,y)=(-15+\Delta x, 36+\Delta y)$$ You know $\Delta y=0.773$. Just plug in the above equation into the equation of the circle: $$(-15+\Delta x)^2=39^2-(36+\Delta y)^2$$ This gives two solutions: $$\Delta x=15\pm\sqrt{39^2-(36+\Delta y)^2}$$ The two pints are symmetric with respect to the middle of the arc.
