I have a line with a circular arc cutting through it at 2 points, A and B.
Given that I know the length of line CD (which is the distance between line AB and the highest point on the arc - perpendicular to AB) and I know the length of line AB, is it possible to calculate the distance between any point on line AB and its corresponding perpendicular point on the arc? for example, if point E is half way between points A and C, how do I calculate the length of EF?
One method can be this:
If $CD=h$ and $AB=b$ then the Radius of the circle is:
$R=\frac{b^2+4h^2}{8h}$
Calculate R.As said in comment extend DC to get $A_1$ such that $DA_1=R$, $A_1$ is the center of the circle. Draw a circle with radius R , center at $A_1$. Draw a line parallel with AB passing $A_1$, it intersects the circle at points G and I, GI is the diameter of the circle. Extend FE to meet GI at J. In right angled triangle FGI we have:
$FJ^2=GI\times JI=(R-d)(R+d)=R^2-d^2$
where d is the distance between CD and EF. Finding FJ we have:
$EF=FJ-(R-h)=FJ+h-R$