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Math Help - HELP. formula for 32 segments of an arc of circle

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    Question HELP. formula for 32 segments of an arc of circle

    I am not a math expert my any means, I have a problem I can not solve.

    I have a circle with a given diameter. I am taking a part of its circumference and dividing it into 32 equal points. I then pull parallel lines from the points and measure the distance between. I understand there are 16 different measurements (mirrored) between these lines. (.p.q=arc/32=.ee.ff) (p,q > ee,ff). What I dont know is what the relationship is between the distances or how to figure the them. I am looking for a formula, as my given diameter and arc distances may change. I hope this is clear. I attached a rough drawing.
    HELP.  formula for 32 segments of an arc of circle-geometry.jpg
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    Re: HELP. formula for 32 segments of an arc of circle

    Quote Originally Posted by Munzer View Post
    I have a circle with a given diameter. I am taking a part of its circumference and dividing it into 32 equal points. I then pull parallel lines from the points and measure the distance between. I understand there are 16 different measurements (mirrored) between these lines. (.p.q=arc/32=.ee.ff) (p,q > ee,ff). What I dont know is what the relationship is between the distances or how to figure the them. I am looking for a formula, as my given diameter and arc distances may change. I hope this is clear. I attached a rough drawing.
    Click image for larger version. 

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    To be quite honest, I am not at all sure what you are doing.
    But have look at this link.
    Formulas 6-9 are for the length of the chord a.
    Now if you need the length of the arc that is formula 2 (in radians).

    If this does help you, then please try to explain what is going o.
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    Re: HELP. formula for 32 segments of an arc of circle

    HELP.  formula for 32 segments of an arc of circle-math.jpg
    I looked over the link you suggest I still am having trouble, I re drew my problem. My questions are:
    what is a? if D=2 and S=2
    what is p? if D=2 amd S=2
    if I change D what is a-p?
    if I change S what is a-p?
    what is the algebraic formula for the relationship of a to b to c etc...

    I believe distance a < p and want to know mathmatically how it is less.
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    Re: HELP. formula for 32 segments of an arc of circle

    Quote Originally Posted by Munzer View Post
    Click image for larger version. 

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    I looked over the link you suggest I still am having trouble, I re drew my problem. My questions are:
    what is a? if D=2 and S=2
    what is p? if D=2 amd S=2
    if I change D what is a-p?
    if I change S what is a-p?
    what is the algebraic formula for the relationship of a to b to c etc...
    I believe distance a < p and want to know mathmatically how it is less.
    Let R=\frac{D}{2}, i.e. the radius,
    If \theta is the radian measure of the central angle subtending the arc S then the length
    S=R\cdot\theta.

    From the new diagram it appears the you the length of the sub-arcs.
    Being equally spaced their length is \frac{S}{32}.
    Now, I am assuming that you mean by a-p the arclength and not the chord length.
    Really all depends upon the measure of the angle and the radius of the circle.
    Last edited by Plato; March 9th 2012 at 01:20 PM.
    Thanks from mash
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  5. #5
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    Re: HELP. formula for 32 segments of an arc of circle

    First, I'm not a mathematician; just a retired grandpa who loves math...
    Code:
    Y(0,r)
    
               C
    
    
                        B
    
    
    
                             A
    
    
     O(0,0)                   X(r,0)
    Much easier to work out by taking a "small case"; let's make it 3 instead of 16;
    we really need to concentrate only on quarter circle, as shown above.

    GIVENS: Arc AY (your S/2), the radius (your D/2), and the number of points (your 16).
    I'm naming them this way:
    k = arc AY : assume 26
    r = radius : assume 20
    n = number of points : 3 as I indicated above

    Calculations (p = pi) :
    g = k/n : ~8.33 (corresponds to your g)
    q = pr / 2 : ~31.42; quarter of circumference (or arc XY)
    t = 90g / q : ~24.83 degrees; angle supporting each g (as example, angle AOB)

    c = rSIN(t) : ~8.40; horizontal distance CY
    b = rSIN(2t) - c : ~6.85; horizontal distance BC
    a = rSIN(3t) - b - c : ~4.03; horizontal distance AB

    I tested (to make sure!) on graph paper and all seems fine.

    However, using n=16 will make it "long" if done as I'm showing,
    so I presume you're also interested in a "quicker" way.
    I'll try and do that in the next post.
    Last edited by Wilmer; March 9th 2012 at 10:34 PM.
    Thanks from mash
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  6. #6
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    Re: HELP. formula for 32 segments of an arc of circle

    OK...in order not to have to assign 16 variables and do 16 calculations,
    I'd suggest you use an array size 16 (or whatever size you need if
    you're trying for something else); make it array A(n), so A(16) if n=16.

    Here's the way it would work (hope you're able to program):

    Get r (radius : assume 1000)
    Get k (full arc : assume 1200)
    Get n (number of points : assume 16)

    Dim A(n) (set up array)

    q = pi * r / 2 (quarter of circumference)
    g = k/n (smaller arcs)
    t = 90 * g / q (size of angle supporting each g)

    For m = 1 to n (loop to calculate each horizontal distance from the y=axis)
    A(m) = r * SIN(t * m) : remember that m increases by 1 each loop
    Next m

    So the array now contains the horizontal distance of each point from the y-axis.
    If you wanted to "see" each along with the distance between consecutive points,
    this command would do it:
    For m = 1 to n
    Print m, A(m), A(m-1)
    Next m
    And you'd see (rounded to 2 places):
    1 74.93 74.93
    2 149.44 74.51
    3 223.11 73.67
    ....
    15 902.27 34.84
    16 932.04 29.77

    And you can get horizontal distance between any 2 points;
    like A(15) - A(3) = 902.27 - 223.11 = 679.16

    Hope that helps. Feel free to question anything I've done...
    Last edited by Wilmer; March 9th 2012 at 10:05 PM.
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