Calculating the dimensions of an arc

[kennethgilpin]

Dear all,

I feel this should be a simple problem but I can't solve it. Could you give me a hand?

Imagine if an arc is bounded by a rectangle of dimensions width and height. The arc starts in the bottom left corner of the rectangle, and ends in the bottom right corner. The apex of the arc is the center of the top side of the rectangle. A picture of this is shown half way down this page:
http://www.mathopenref.com/chordsintersecting.html"

Assuming that the arc is less than a semi circle, if I know the length of the arc and the height of the rectangle, how can I calculate it's width?

Any help gratefully received,

kenneth

 

[micromass]

Hi kennethgilpin!

So you know some calculus? In my opinion that's the easiest way to solve this.

First we will have to find the equation of the circle given the coordinates of the rectangle.
Say that the vertices of the rectangle have coordinates (0,L),(0,-L),(H,L),(H,-L).

The general equation of a circle is

$$(x-x_0)^2+(y-y_0)^2=R^2$$

with (x₀,y₀) the center of the circle and R the radius. We see easily that the center of the circle must lie on the y-axis, and thus x₀=0.

Now, what you have to do is to find y₀ and R such that (0,L) and (H,0) lie on the circle

$$x^2+(y-y_0)^2=R^2$$

 

[I like Serena]

@MM: I'm afraid this won't work.
It's the arc length that is given, and the width that is asked.

I believe that the resulting equations can not be solved algebraically.
Of course it can be approximated numerically.

 

[micromass]

@MM: I'm afraid this won't work.
It's the arc length that is given, and the width that is asked.

Yes, I know. What I was attempting is to find an equation which calculated the arc length if you have the length and width given. Then I would use this equation to find an equation calculating the width... I think this ought to work, given that the equations are not too difficult...

 

[I like Serena]

I created the set of equations:

$$\begin{eqnarray}<br /> S &=& 2 R \alpha \\<br /> W &=& 2 R \sin \alpha \\<br /> R &=& H + R \cos \alpha<br /> \end{eqnarray}$$

where S is the arc length, and $\alpha$ is half of the angle of the arc.

If you try to solve it for W, with given S and H, you'll quickly see that you're left with an equation that is afaik not solvable algebraically.

 

[kennethgilpin]

thank you for your replies. @likeSerena - i got similar equations to you which I could not solve.

This is for use within an iterative model so I will try to solve it iteratively.

kenneth