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Q: Mathematical Formula Sought Relating to Radius of Circle, Arc Length and Deflect ( Answered ,   2 Comments )
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 Subject: Mathematical Formula Sought Relating to Radius of Circle, Arc Length and Deflect Category: Science > Math Asked by: fingersfinny-ga List Price: \$5.00 Posted: 19 Oct 2004 16:13 PDT Expires: 18 Nov 2004 15:13 PST Question ID: 417212
 ```Consider an Arc of a circle, arc length = L. The ends of the Arc are points A and B respectively. If a line was struck between A and B - at the mid point of this line, the distance to the Arc (perpendicular to AB) is Delta. My question is - If L is known and Delta is known (Length AB is not known) what is the formula to be used to determine the radius of the circle. NO APPROXIMATIONS. I am looking for a formula beginning R=.``` Request for Question Clarification by mathtalk-ga on 19 Oct 2004 20:34 PDT ```Hi, fingersfinny-ga: The exact value of R can only be expressed in terms of a function that is unlikely to be familiar to you. Let me explain why. You are given the length L and the "chord height" delta or D. Provided that 0 < D * pi < L, the radius R and the central angle T which subtends the arc can be uniquely determined. Provided angle T is given in radians, the two relations which must be solved are these: R * T = L R * (1 - cos(T/2)) = D If we are able to devise a forumla for R in terms of L and D, then the first equation would give us a formula for T, i.e. T = L/R. Conversely if we are able to express T by a formula, then R = L/T. The easiest first step is to eliminate R from the two equations. Since the sides of both equations are positive, we can divide the left-hand sides and set that equal to the ratio of the right-hand sides: (1 - cos(T/2))/T = D/L The crux of the problem of finding a "formula" to express R or T is here. The left-hand side depends only on T, and the right-hand side is a known value. The unique value of T between 0 and 2pi that satisfies this is the solution of a transcendental equation. One can explicitly find the power series in T that represents the left-hand side, and then the problem becomes one of "inverting" that power series. The resulting expression will be exact, but it will not be one of the familiar power series functions, like arctangent for example. If you are interested in such formulas for inverting a power series function like (1 - cos(T/2))/T, let me know whether such an Answer would be acceptable. regards, mathtalk-ga``` Clarification of Question by fingersfinny-ga on 20 Oct 2004 12:26 PDT ```You state "the easiest first step is to eliminate R" - are you suggesting that there is another route (not-so-easy) which retains R such that you might end up with a formulae / equation commencing R=? Does it make any difference if the solution is only valid for angle T is less than 180 degrees/Pi Radians? Such that a general solution is not required for all T.``` Request for Question Clarification by mathtalk-ga on 20 Oct 2004 19:10 PDT ```No, it doesn't make it any easier to give a formula exactly valid for a smaller range of angles. The solution is that R = L/T where T is a convergent power series in D/L. Functions like this are not "broken up" into separate definitions on subdomains. Their definitions are "all of one piece" in a sense made precise by analytic function theory. Whatever route you take to an exact solution, you will wind up with the same solution. It will not be polynomial or a rational function of L and D, nor will it be expressed as a finite combination of the familiar elementary transcendental functions (sin, cos, tan, log, exp, etc.), though in principle the definition of the solution is really no better or worse than the definition of any of these more familiar functions. regards, mathtalk-ga``` Clarification of Question by fingersfinny-ga on 25 Oct 2004 17:18 PDT `Mathtalk - please post your answer as the official answer. Many thanks.`
 Subject: Re: Mathematical Formula Sought Relating to Radius of Circle, Arc Length and Def Answered By: mathtalk-ga on 31 Oct 2004 14:14 PST Rated:
 ```Hi, fingersfinny-ga: Thanks for inviting me to post an "official" Answer. I'm afraid that I have written far more below than you really wanted to know about radius of a circle from arc length and "deflection", so let me summarize the main points here, and you can pick through as much of what follows as seems interesting. I'll recap the approach outlined before, solving a pair of equations for unknown radius and central angle from given values of arc length and deflection (chord height). One point not discussed before is that solutions are not unique unless the size of the central angle is suitably restricted. That is, for each ratio of deflection to arc length, there are at least two central angles which correspond to that ratio. There's no ambiguity, however, if the problem involves arcs that are no bigger than a semicircle. At about 4.6 radians, the ratio of deflection to arc length peaks out & begins to decrease with increasing an central angle. The maximum ratio that can be attained is about 0.36. Therefore we will restrict our efforts to solve the problem to positive ratios less than this, and so to central angles between 0 and roughly 4.6 radians. Outline of Solution =================== A circle of unknown radius R has a chord that cuts off an arc of length L, and the "deflection" or chord height (perpendicular distance from midpoint of chord to midpoint of arc) is D or delta. By introducing an unknown central angle T subtended by the chord (or by the arc), we can formulate two equations with two unknowns: L = T * R D = R - R * cos(T/2) The first equation says arc length L is the product of central angle T (measured in radians) and radius R. Applied to the entire circle this gives that the circumference is 2pi * R. Naturally this requires equal units of measurement for lengths L and R. The second equation expresses the deflection D as a difference between a radius R through the midpoint of the chord and the included leg of a right triangle, drawn to that midpoint from the center of the circle (where an endpoint of the chord is the third vertex). Note T/2 is half the central angle and that half the length of chord (the other leg of this right triangle) is R * sin(T/2). We propose ultimately solving by first taking the ratios of these two equations' respective sides, which eliminates R: D/L = (1 - cos(T/2))/T (*) Solving this nonlinear equation for central angle T and plugging back into our first equation would give the radius: R = L/T Note that radius R is proportional to L and inversely proportional to central angle T. It's often recommended that (*) be solved numerically, and in practice this is easily accomplished by any of the basic root finding methods. Cf. the "Dr. Math" links posted for you by mattstephens30-ga below. To solving it symbolically, on the other hand, will give insight into precisely how the solution for T varies with D/L. What we need is an inverse function to (1 - cos(T/2))/T, ie. a function f that "undoes" what this expression does and gives us back T: f( (1 - cos(T/2))/T ) = T Armed with this function we could then write from (*): f( D/L ) = T and R = L / f(D/L) * * * * * * * * * * * * * * * * * * Illustration ============ Let's interrupt the narrative at this point with an example, which almost always helps to clear up any ambiguity about the notation. Since all the distances are relative to some unit of measurement, one can assume without loss of generality that the arc length L = 1 where of course the units of radius R will scale proportionately. In other words only the ratio D/L matters, and for the sake of illustration we take D = 0.1 and L = 1.0. T is then f(0.1), ie. the central angle such that: (1 - cos(T/2))/T = D/L = 0.1 The slope of the left-hand side as a function of T turns out below to be 1/8, so a first-order approximation to T would be 8 * 0.1. In fact: (1 - cos(0.4))/0.8 = 0.09867... which isn't too far off. Refinement of this estimate leads to: f(0.1) ~ 0.811055 The corresponding radius would then be R = 1/T ~ 1.232962. As mentioned before, the solutions are only unique if we restrict our attention to angles T in roughly (0,4.6), which corresponds to ratios D/L between 0 and about 0.36. So with f(0.1) we have already gone a substantial way to the outer limit of what parameters can be solved. * * * * * * * * * * * * * * * * * * Symbolic Solutions ================== Let's resume analysis with the familiar power series for cosine: +oo cos(x) = 1 + SUM (-1)^k x^(2k) / (2k)! k=1 = 1 - x^2/2 + x^4/12 - x^6/720 + x^8/40320 - ... Apply this expression for cos(T/2) and simplify: 1 - cos(T/2) +oo ------------ = SUM (1/2) (-1)^(k-1) (T/2)^(2k-1) / (2k)! T k=1 3 5 7 9 T T T T T = - - --- + ----- - -------- + ---------- + ... 8 384 46080 10321920 3715891200 Inverting a power series (analytic) expansion about T = 0 is possible theoretically whenever the derivative is nonzero: [Inverse Function Theorem -- PlanetMath] http://planetmath.org/encyclopedia/InverseFunctionTheorem.html Note that our series derivative at T = 0 is 1/8, which is nonzero, and conveniently the constant term is zero, so the series is zero at T = 0 and we can thus expect to define an inverse function with f(0) = 0. In fact there's a theorem due to Lagrange which gives a sort of recipe for the power series of this sort of inverse function: [Lagrange Inversion Theorem -- Wikipedia] http://en.wikipedia.org/wiki/Lagrange_inversion_theorem [Series Reversion -- Eric Weisstein (MathWorld--A Wolfram Web Resource)] http://mathworld.wolfram.com/SeriesReversion.html It's not too hard to work out "by hand" the first several terms of such a power series for an inverse function, but beyond that things get tough. I've resorted to an open source symbolic algebra package called Maxima: [Maxima - a sophisticated computer algebra system] http://maxima.sourceforge.net/ Let d = D/L be the "dimensionless" ratio on the left-hand side of (*), so that for clarity we can write f(d) = T. Thus 3 5 7 9 32 d 1664 d 159232 d 4139008 d T = f(d) = 8 d + ----- + ------- + --------- + ---------- + . . . 3 45 945 4725 A convergent power series expansion, carried to infinity, gives exactly the central angle T. However convergence is only guaranteed for "small" d = D/L. A graph of (1 - cos(T/2))/T will show that it oscillates up and down. No single-valued inverse can be defined globally, and series expansions around d = 0 can only converge in a symmetric interval about the origin if they exclude the nearest singularity. Moreover the convergence of a power series is progressively slower as we move away from d = 0. One way to think about this slow convergence is that we have a sequence of polynomials (power series convergents) that "fit" the value of f(d) and successively higher derivatives at the origin. The power series approximation is therefore optimal near d = 0 but apt to be unsuitable at modest distances from the origin. Faster convergence is often obtained with rational approximations, and in the context of exact expressions for the inverse, this leads to the computation of a continued fraction formula rather than a power series. Just as a power series can be expanded about any point interior to the domain of the function, there is flexibility in constructing continued fraction expansions. Unfortunately much of the discussion of continued fractions of the Web dwells on best rational approximations to individual real numbers. A brief introduction to continued fraction function expansions is tacked on to end here: [Continued Fractions -- Numericana] http://home.att.net/~numericana/answer/fractions.htm As the thread in this forum mentions: [Continued Fractions -- Ask NRICH] http://nrich.maths.org/discus/messages/2069/6603.html?1064175671 "...[t]here is a general scrappy way of getting a continued fraction expansion from the power series..." which we will illustrate upon our example above. Begin by factoring out the first term 8d, leaving an even series: 2 4 6 8 4 d 208 d 19904 d 517376 d f(d) = 8d * ( 1 + ---- + ------ + -------- + --------- + ... ) 3 45 945 4725 Now replace multiplication by this even series with division by its reciprocal series (which exists because its constant term is a unit): 2 4 6 8 4 d 128 d 10496 d 151552 d f(d) = 8d / ( 1 - ---- - ------ - -------- - --------- - ... ) 3 45 945 2835 Continue in this vein, factoring out -dē from each of the higher order terms in the denominator: 2 4 6 2 4 128 d 10496 d 151552 d f(d) = 8d/( 1 - d *( - + ------ + -------- + --------- + ... ) ) 3 45 945 2835 and again rewrite the product as division by a reciprocal series: 2 4 6 2 3 8 d 496 d 12032 d f(d) = 8d/(1 - d /( - - ---- - ------ - --------- + ... ) ) 4 5 175 1125 Grind the crank a few more times in this fashion: 2 4 2 3 2 5 31 d 1955 d f(d) = 8d/(1 - d /( - - d /( - - ----- - ------- + ... ) ) ) 4 8 28 882 and: 2 2 3 2 5 2 28 15640 d f(d) = 8d/(1 - d /( - - d /( - - d /( -- - ------- + ... ) ) ) 4 8 31 8649 2 3 2 5 2 28 2 8649 = 8d/(1 - d /( - - d /( - - d /( -- - d /( ----- - ... ) ) ) ) 4 8 31 15640 Until the appearance of this last "ugly" constant in the continued fraction conversion, it certainly seemed that a simpler expression was emerging. I suspect that with further exploration of various alternatives, a formula with an explicit pattern of constants can be discovered. * * * * * * * * * * * * * * * * * * Convergence of Power Series vs. Continued Fraction ================================================== Let's finish by returning briefly to the numeric illustration shown earlier, where d = D/L = 0.1, and compare how well the power series solution converges there versus the continued fraction expansion. First the results using n terms of the power series expansion: n approx. value of T --- ------------------ 1 0.8 2 0.81066666666667 3 0.81103644444444 4 0.81105329439153 5 0.81105417037206 6 0.81105421959762 7 0.81105422250983 8 0.81105422268853 Because the series expansion contains only odd powers, n terms will correspond to retaining highest power 2n-1. Thus the values shown preserve greater accuracy in the bottom rows than we retained above in our discussion. Now let's present similar results extracting n continued fraction terms from the fullest power series derived above: n approx. value of T --- ------------------ 1 0.8 2 0.81081081081081 3 0.81104972375691 4 0.81105412922206 5 0.8110542209176 6 0.81105422266419 7 0.81105422269988 8 0.81105422272161 Even though the power series above is converging well for d = 0.1, and despite taking the continued fraction expansion from the that of the power series (instead of computing one independently), the approximations from the continued fraction are better at each step. Indeed n = 5,6,7 terms of the continued fraction are better than n = 6,7,8 terms, respectively, in the power series. The superiority of the continued fraction becomes more dramatic as the value of d increases. For example at d = 0.3, just 5 terms in the continued fraction give a more accurate solution than 8 terms in the power series. regards, mathtalk-ga```
 fingersfinny-ga rated this answer:

 ```All explained in: http://mathforum.org/dr.math/faq/faq.circle.segment.html There is also an accompanying excel spreadsheet that does the calculation. http://mathforum.org/dr.math/gifs/ChordMath.xls If you open the spreadsheet you have h = delta and s = L Enter these in the yellow boxes and check out case 3. Hope this helps```
 ```Hi, fingersfinny-ga: Thanks, I have a bit of algebra to do to put an Answer in final form for you. regards, mathtalk-ga```