I have search the web for an answer. Here is what I know. I know the radius (500") of the arc. I know the arc height(48") from the center of the chord. Starting at the center of the cord find the length to the arc every 15 degrees. If the formula to find the answer does not input values provided more than likely I will not be able to understand.
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So you have a circle with radius 500 units and a chord perpendicular to a radius which crosses the radius a distance 452 units from the center of the circle and you wish to know the distance from the center of the chord to the arc of the chord as a function of the angle between the chord and the segment connecting the center of the chord to the arc? – John Wayland Bales Jul 10 '16 at 19:21
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That is correct. After looking at your diagram it appears to be what I am trying to figure out. I will try to work your formula to see if it is what I am looking for. I am checking my answers using a CAD program. Thank you for your quick response. – James Jul 12 '16 at 16:02
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The point $C$ has coordinates $(0,452)$. In order to find the length of segment $CG$ as a function of the angle $\alpha$ one can first find the coordinates of the point $G$ as functions of $\alpha$. The point $G$ is one of the two points of intersection (the one with positive $x$-coordinate) of the two graphs:
\begin{equation} y=x\tan\alpha+452 \end{equation}
\begin{equation} x^2+y^2=500^2 \end{equation}
Substituting the first equation into the second gives
\begin{equation} x^2+(x\tan\alpha+452)^2=500 \end{equation}
which can be solved for the non-negative value of $x$ in terms of $\alpha$.
Then using the first of the two equations above, one can find $y$ as a function of $\alpha$.
Then one can derive the length of $CG$ as a function of $\alpha$ for $0\le\alpha\le\tfrac{\pi}{2}$. For $\tfrac{\pi}{2}<\alpha\le\pi$ the distance is the same as for the acute angle $\pi-\alpha$.
John Wayland Bales
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Again I would like to think you for the formula you have provided. I am unable to find the length of CG. Please step through your formula until the answer is given. I have tried to understand what I need to do by going to a few other sites but all do not give specific examples that I can figure out that apply to what I am looking for. Your formula appears elegant but is still beyond my capabilities. – James Jul 12 '16 at 17:28
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I believe that this is a problem which would be easier to work in a general rather than a specific case. What I mean is use a unit circle $x^2+y^2=1$ and a horizontal line $y=b$. Then find the lengths from the center of the chord to the arc as a function of $\alpha$ in this simpler setting. Then to solve the case when the circle radius is $500$ and the 'height' of the chord is $48$ you would let $b=\dfrac{500-48}{500}=0.904$ and multiply the resulting distances by $500$ to get the distances for a circle of radius $500$. I will take a look at that and see what I can come up with. – John Wayland Bales Jul 12 '16 at 18:30
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I find that for the circle $x^2+y^2=1$ and the horizontal chord lying on $y=b$ where $0\le b<1$ the length of $CG$ in the diagram is $\dfrac{\sqrt{\sec^2\alpha-b^2}-b\tan\alpha}{\sec\alpha}$. In your case you would use $b=0.904$ and multiply the result by $500$ to get $|OG|=500\cdot\dfrac{\sqrt{\sec^2\alpha-0.904^2}-(0.904)\tan\alpha}{\sec\alpha}$ for acute angles $\alpha$. For obtuse angles $\alpha$, $|OG|$ would be the same distance as for the acute angle $180^\circ-\alpha$. – John Wayland Bales Jul 12 '16 at 19:08
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When I wrote $OG$ I meant $CG$. But for which angle $\alpha$ are you computing $CG$? The value will be different for each angle from $0^\circ$ to $90^\circ$. – John Wayland Bales Jul 12 '16 at 19:31
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See original post for a Cad drawing of the answer I am trying to find a formula for. Click on "Answer I am Looking for". Thank you for your continued help. – James Jul 12 '16 at 19:32
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Yes, that is the correct value for $15^\circ$. After computing it, since there is no secant function on my calculator I see a better formula can be had by multiplying numerator and denominator by $\cos\alpha$ to get $|CG|=500\left[\sqrt{1-(0.904\cos\alpha)^2}-(0.904)\sin\alpha\right]$. – John Wayland Bales Jul 12 '16 at 19:36
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Note that the formula $|CG|=500\left[\sqrt{1-(0.904\cos\alpha)^2}-(0.904)\sin\alpha\right]$ also gives the result 213.7662 when $\alpha=0$ which agrees with your CAD diagram so I am confident that the formula is correct. You should be able to use it to find the correct distance for other angles if necessary. – John Wayland Bales Jul 12 '16 at 19:43
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The most general formula for an arc of radius $R$ and height $H$ would be $$\text{LENGTH }=\sqrt{R^2-[(R-H)\cos\alpha]^2}-(R-H)\sin\alpha$$ in case you ever meet a similar problem – John Wayland Bales Jul 12 '16 at 19:56