IEEE TRANACTION ON INTRUMENTATION AND MEAUREMENT, VOL. XX, NO. Y, MONTH 000 A Few Methods for Fittig Circles to Data Dale Umbach, Kerry N. Joes Abstract Five methods are discussed to fit circles to data. Two of the methods are show to be highly sesitive to measuremet error. The other three are show to be quite stable i this regard. Of the stable methods, two have the advatage of havig closed form solutios. A positive aspect of all of these models is that they are coordiate free i the sese that the same estimatig circles are produced o matter where the axes of the coordiate system are located or how they are orieted. A atural extesio to fittig spheres to poits i 3-space is also give. Key Words: Least quares, Fittig Circles AM ubject Classificatio: 93E4 I. Itroductio THE problem of fittig a circle to a collectio of poits i the plae is a fairly ew oe. I particular, it is a importat problem i metrology ad microwave measuremet. While certaily ot the earliest referece to a problem of this type, Kȧsa, i [4], describes a circle fittig procedure. I [], Cox ad Joes expad o this idea to fit circles based o a more geeral error structure. I geeral, suppose that we have a collectio of 3 poits i -space labeled x, y, x, y,..., x, y. Our basic problem is to fid a circle that best represets the data i some sese. With our circle described by x a + y b = r, we eed to determie values for the ceter a, b ad the radius r for the best fittig circle. A reasoable measure of the fit of the circle x a + y b = r to the poits x, y, x, y,..., x, y is give by summig the squares of the distaces from the poits to the circle. This measure is give by a, b, r = r x i a + y i b [] discusses umerical algorithms for the miimizatio over a, b, ad r. Gader, Golub, ad trebel i [3] also discuss this problem. I [4], Kȧsa also presets a alterative method that we will discuss i ectio.4. [] gives a slight geeralizatio of the Kȧsa method. II. The Various Methods For otatioal coveiece, we make the followig covetios: X ij = x i x j X ijk = X ij X jk X ki D. Umbach ad K. N. Joes are with the Departmet of Mathematical cieces, Ball tate Uiversity, Mucie, IN 47306, UA. E-mail: dumbach@bsu.edu ad kjoes@bsu.edu X ij = x i x j Y ij = y j y j Ỹ ijk = Y ij Y jk Y ki Y ij = y i y j A. Full Least quares Method A obvious approach is to choose a, b, ad r to miimize. Differetiatio of yields r a b = = r = r xi a + y i b +r x + a x i a xi a + y i b y i b xi a + y i b y + b. II. imultaeously equatig these partials to zero does ot produce closed form solutios for a, b, ad r. However, may software programs will umerically carry out this process quite efficietly. We shall refer to this method as the Full Least quares method FL with resultig values of a, b, ad r labeled as a F, b F, ad r F. The calculatio of the FL estimates has bee discussed i [] ad [3], amog others. B. Average of Itersectios Method We ote that solvig II. = 0 for r produces r = xi a + y i b /. II. This suggests that if oe obtais values of a ad b by some other method, a good value for r ca be obtaied usig II.. To obtai a value for a, b, the ceter of the circle, we ote that for a circle the perpedicular bisectors of all chords itersect at the ceter. There are 3 triplets of poits that could each be cosidered as edpoits of chords alog the circle. Each of these triplets would thus produce a estimate for the ceter. Thus, oe could average all 3
IEEE TRANACTION ON INTRUMENTATION AND MEAUREMENT, VOL. XX, NO. Y, MONTH 000 of these estimates to obtai a value for the ceter. Usig these values i II. the produces a value for the radius. We shall refer to this method as the Average of Itersectios Method AI with resultig values of a, b, ad r labeled as a A, b A, ad r A. A positive aspect of this method is that it yields closed form solutios. I particular, with we have w ijk = x i Y jk + x j Y ki + x k Y ij II.3 a A = w ijk = x i Y jk + x jy ki + x ky ij z ijk = y i X jk + y j X ki + y k X ij II.4 b A = r A = z ijk = y i X jk + y j X ki + y kx ij, w ijk Ỹijk w 3 j=i+ ijk k=j+ z ijk X ijk z 3 j=i+ ijk k=j+ xi a I + y i b I / A obvious drawback to this method is that it fails if ay three of the poits are colliear. This is obvious from the costructio, but it also follows from the fact that if x i, y i, x j, y j, ad x k, y k are colliear the w ijk = 0 ad z ijk = 0 i II.3 ad II.4. The method is also very ustable i that small chages i relatively close poits ca drastically chage some of the approximatig ceters, thus producig very differet circles. This method is similar to fittig a circle to each of the triplets of poits, thus gettig 3 estimates of the coordiates of the ceter ad the radius, ad the averagig these results for each of the three parameters. It differs i the calculatio of the radius. AI averages the distace from each of the poits to the same ceter a I, b I. C. Reduced Least quares Method This leads to cosideratio of differet estimates of the ceter a, b. Agai, if all of the data poits lie o a circle the the perpedicular bisectors of the lie segmets coectig them will itersect at the same poit, amely a, b. Thus it seems reasoable to locate the ceter of the circle at the poit where the sum of the distaces from a, b to each of the perpedicular bisectors is miimum. Thus, we seek to miimize Ra, b = j=i+ ax + by 0.5Y + X X + Y II.5 As i the Full Least quares method, equatig the partial derivatives of R to zero does ot produce closed form solutios for a ad b. Agai, however, umerical solutios are ot difficult. Let us label the resultig values for a ad b as a R ad b R. Usig these solutios i II. yields the radius of the fitted circle, r R. We shall refer to this method as the Reduced Least quares method RL. As will be discussed i ectio 3, this method of estimatio is ot very stable. I particular, the X + Y i the deomiator of II.5 becomes problematic whe two data poits are very close together. D. Modified Least quares Methods To dowweight pairs of poits that are close together, we will cosider miimizatio of Ma, b = ax + by 0.5X ij + Y ij j=i+ Differetiatio of M yields M a M b = b j=i+ +a = a j=i+ j=i+ +b j=i+ X Y X j=i+ j=i+ Y X Y j=i+ j=i+ We ote that for ay vectors α i ad β i, j=i+ α j α i β j β i = X Y X X Y X Y Y α i β i α i β i II.6 II.7 Notig that II.7 is αβ, where αβ is the usual covariace, we see that equatig these partial derivatives to zero produces a pair of liear equatios whose solutio ca be expressed as where A = B = a M = b M = DC BE AC B AE BD AC B, x i x i = x x i y i x i y i II.8 II.9 II.0
IEEE TRANACTION ON INTRUMENTATION AND MEAUREMENT, VOL. XX, NO. Y, MONTH 000 3 = xy II. C = yi y i = y II. D = 0.5 { + x i yi x i x 3 i x i x i y i } = 0.5 xy + xx II.3 { E = 0.5 y i x i y i + yi 3 y i y i x i } = 0.5 yx + yy II.4 Agai, we fid the radius usig II. as r M = xi a M + y i b M / II.5 We shall refer to this as the Modified Least quares method ML. A differet approach was preseted i [4]. There, Kȧsa proposes choosig a, b, ad r to miimize Ka, b, r = r x i a y i b He idicates that solutio for a ad b ca be obtaied by solvig liear equatios, but does ot describe the result of the process much further. It ca be show that the miimizatio of K produces the same ceter for the fitted circle as the ML method. The miimizig value of r, say r K, is slightly differet from r M. It turs out that r K = x i a M + y i b M / By Jese s iequality, we see that r K, beig the square root of the average of squares, is at least as large as r M, beig the correspodig average. III. Compariso of the Methods We first ote that if the data poits all truly lie o a circle with ceter a, b ad radius r, the all five methods will produce this circle. For FL, this follows sice the oegative fuctio is 0 at a, b, r. For AI, this follows from the observatio that the itersectios of all of the perpedicular bisectors occur at the same poit a, b, ad hece each of the values i II. that are to be averaged is r, ad hece r A = r. For RL ad ML, we ote that the terms i M i II.6 are all 0 at a, b as the are the terms i R as well. Agai, the radii of.5 0.5 0.5.5 ML,FL, Kasa Fig.. Fits of FL, AI, RL, ML, ad Kȧsa circles to five data poits. the RL ad ML methods are r for the same reaso as give for AI. ice the values to be averaged for r M are all idetical, we also have r K = r M = r. If ay three poits are colliear, the AI fails because the perpedicular bisectors for this triple are parallel, thus producig o itersectio poit. Thus averagig over the itersectio poits of all triples fails. The other four methods produce uique results i this situatio, uless, of course, all of the data poits are colliear. If all of the poits are colliear, the all five methods fail. FL fails because the larger the radius of the circle, with appropriate chage i the ceter, the closer the fit to the data. For ML ad Kȧsa, we ote that if all of the data poits are colliear, the xy = x y, ad hece the deomiators of a M ad b M i II.8 ad II.9 are 0 usig II.0, II., ad II.. RL fails because for this case there will be a ifiite collectio of poits that miimize the distace from the poit a, b to the parallel lies that form the perpedicular bisectors. To give a idicatio of how sesitive the methods are to measuremet error, we cosider fits to a few collectios of data. All five methods fit the circle x +y = to the followig collectio of five poits, 0,,,,,0,,, ad 0.05, + 0.09775. uppose that the last data poit, however, was icorrectly recorded as 0.03,.0, a poit oly 0.539 uits away. The results of the fits to these five poits are displayed i Figure. As is evidet from the figure, we see that the AI circle was drastically affected. The fit is ot close at all to the circle of radius cetered at,. Not quite as drastically affected, but seriously affected, oetheless, is the RL circle. I cotrast, the FL, Kȧsa, ad ML circles are ot perspectively differet from the circle of radius cetered at,. This strogly suggests that the FL, Kȧsa, ad ML methods are robust agaist measuremet error. Figure cotais FL, ML, ad Kȧsa fits to the followig seve data poits, 0,,,,,.5,.5,0, 0.5,0.7, 0.5,, ad.5,.. These three circles are fairly similar, but ot idetical. Each seems to describe the data poits well. It is ope to iterpretatio as to which circle best fits the seve poits. These fits poit favorably to usig the ML ad Kȧsa RL AI
IEEE TRANACTION ON INTRUMENTATION AND MEAUREMENT, VOL. XX, NO. Y, MONTH 000 4.5 ML Kasa FL Miimizatio of R requires a umerical solutio. This solutio also suffers i that it is very sesitive to chages i data poits that are close together. Thus, as i the dimesioal case, we cosider the modificatio produced by istead miimizig M a, b, c = j=i+ ax + by + cz 0.5X + Y + Z 0.5 Fig.. 0.5.5 Fits of FL, ML, ad Kȧsa circles to seve data poits. methods to fit circles. The robustess of the methods ad the existece of closed form solutios are very appealig properties. Recall that these circles are cocetric, with the Kȧsa circle outside the ML circle. Thus, outliers iside the circles would make the Kȧsa fit seem superior. Whereas, outliers outside the circles would make the ML fit seem superior. IV. Fittig pheres i 3-space These methods are ot difficult to geeralize to fittig spheres to poits i 3-space. o, suppose that we have a collectio of 4 poits i 3-space labeled x, y, z, x, y, z,..., x, y, z. The basic problem is to fid a sphere that best represets the data i some sese. With our sphere described by x a + y b + z c = r, we eed to determie values for the ceter a, b, c ad the radius r for the best fittig circle. Based o the comparative results i ectio 3, we will oly cosider extesios of the FL, ML, ad Kȧsa methods. For FL, we seek to miimize a, b, r = r x i a + y i b + z i c As i the dimesioal case, oe must resort to umerical solutios. The derivatio of the ML estimate proceeds i a similar maer i 3-space. The plae passig through the midpoit of ay chord of a sphere which is perpedicular to that chord will pass through the ceter of the sphere. Thus we seek the poit a, b, c which miimizes the sum of the squares of the distaces from a, b, c to each of the plaes formed by pairs of poits. This leads to miimizatio of R a, b, c = j=i+ X a + Y b + Z c 0.5X + Y + Z X + Y + Z Aalogous to the dimesioal case, we obtai closed form solutios, a M, b M, c M, to the miimizatio problem. Defiig the mea squares as i II.0 through II.4, we obtai a M = b M = c M = xx + xy + xz y z yz + yx + yy + yz xz yz xy z + zx + zy + zz xy yz xz y x y z + xy yz xz x yz y xz z xy xx + xy + xz xz yz xy z + yx + yy + yz x z xz + zx + zy + zz xy xz yz x x y z + xy yz xz x yz y xz z xy xx + xy + xz xy yz xy y + yx + yy + yz xz xy yz x + zx + zy + zz x y xy Aalogous to II.5, we fid r M = x y z + xy yz xz x yz y xz z xy xi a M + y i b M + z i c M / For this problem, it is ot difficult to show that the fit of [4] has the ceter described by a M, b M, ad c M. The radius for the fitted circle is r K = x i a M + y i b M + z i c M / Refereces [] I.D. Coope, Circle fittig by liear ad oliear least squares Joural of Optimizatio Theory ad Applicatios vol. 76, pp. 38-388, 993. [] M.G. Cox ad H.M. Joes, A algorithm for least squares circle fittig to data with specified ucertaity ellipses IMA Joural of Numerical Aalysis vol. 9, pp. 85-98, 989. [3] W. Gader, G.H. Golub ad R. trebel, Least squares fittig of circles ad ellipses Bulleti of the Belgia Mathematical ociety vol. 3, pp. 63-84, 996. [4] I. Kȧsa, A circle fittig procedure ad its error aalysis IEEE Trasactios o Istrumetatio ad Measuremet vol. 5, pp. 8-4, 976.
IEEE TRANACTION ON INTRUMENTATION AND MEAUREMENT, VOL. XX, NO. Y, MONTH 000 5 Dale Umbach received his Ph.D. i statistics from Iowa tate Uiversity. He was a assistat professor at the Uiversity of Oklahoma for a short time. ice the, he has bee o the faculty of Ball tate Uiversity teachig mathematics ad statistics. He is curretly servig as the chair of the Departmet of Mathematical cieces. Kerry Joes is a geometric topologist specializig i 3-dimesioal maifolds. ice 993, he has bee o the faculty of Ball tate Uiversity, where he is curretly Associate Professor ad Assistat Chair of the Departmet of Mathematical cieces. He has also served o the faculties of the Uiversity of Texas ad Rice Uiversity, where he received his Ph.D. i 990. I additio, he serves as Chief Techical Officer for Pocket oft, a Housto-based software firm ad was formerly a egieer i the ystem Desig ad Aalysis group at E-ystems ow Raytheo i Dallas, Texas.