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The phase space acceptance of a helical quadrupole channel of finite length Ohnuma, Shoroku Oct 31, 1969

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T R 1-69-10TRIUMFTHE PHASE SPACE ACCEPTANCE OF A HELICAL QUADRUPOLE CHANNEL OF FINITE LENGTHby Shoroku OhnumaVisiting from Yale UniversityPhysics Department University of Victoria October 1969ABSTRACTThe acceptance region in the four-dimensional phase space of a helical quadrupole channel has been investigated numerically. The two transverse directions are strongly coupled and the projection on the (x, x') plane of the four-dimensional acceptance region is not independent of (y, y')- If such a channel is used as a pion or a muon channel, it has to be matched to quadrupoles and bending magnets so that its effective acceptance would become smaller than the acceptance region in the four-dimensional phase space.In calculating the acceptance, a circular aperture has been assumed. Also, the axial magnetic field Bz is assumed to be small so that the transverse equations of motion are linear.INTRODUCTIONA nossibility cf using a helical quadrupole as a muon channel has recentlybeen discussed by R.M. Pearce'. By numerically integrating the equationsof motion, he showed that, for a point source on the axis, the helicalchannel is stronger in the focusing action than an a 1 ternating-gradientchannel. Since the construction of such a helical quadrupole seems tobe technically feasible and the cost could be considerably less than that2of an ordinary AG channel , it is interesting to investigate its accept­ance in detail and to compare with the performance of a conventional AG3muon channel .The overall acceptance A(x, x 1, y, y 1) of an infinitely long helical quadrupole in the four-dimensional phase space (x, dx/dz, y, dy/dz) has already been calculated by Salardi et al. for the case where the period A of the pole tip rotation is much larger than the aperture R so that the axial component of the magnetic field, , can be neglected. The acceptance A(x, x 1, y, y 1) for a given aperture R is proportional to A~2 . For a reasonable choice of A/R = 20, the four-dimensional acceptance of a helical quadrupole for the optimum momentum is about202; larger than the corresponding acceptance of AG channels'*. Onedefinite disadvantage of a helical channel is its low acceptance for momenta smaller than the optimum momentum, p(optimum). The lowest acceptable momentum is about 0 .8p(optimum) compared to 0.6 to 0.7 p(optimum) for an AG channel. The normalized four-dimensional accept­ance calculated by Salardi, A(x, x 1 , y, y 1 )A2/RI+ is shown in Fig. 1 as- 3 -a function of p/p (optimum). For a muon channel, it is usually desirable to choose channel parameters such that the acceptance for the pion momentum, p , and for the muon momentum, p , is the same. Fiq. 27T Uindicates the decrease of this acceptance as the ratio p /p deviatesIT pfrom unity.The most important difference of a helical quadrupole from AG channels (and, at least in the first order in x, x', y and y 1, from bending magnets) is the strong coupling of two transverse directions, x and y. One cannot discuss the two-dimensional acceptance in (x, x 1) phase space independent of (y, y 1). This situation is of course the same for solenoid channels^ and for coaxial channels^. When these channels are used together with ordinary quadrupoles and bending magnets, as is usually the case, perfect matching of acceptances is impossible and the acceptance of a helical quadrupole (or of a solenoid) diminishes effect ively.In the following sections, computer programs for studying the acceptance of a helical channel of a finite length are described together with some results. In particular, as a reasonable measure of the effective two- dimensional acceptance, an acceptance area in (x, x 1) phase space common to all accepted values of y and y' can be found by these programs. Throughout the following discussion, the axial magnetic field is neglected and notations are the same (unless specifically explained) as in RMP^. For the sake of convenience, notations used in the article by Salardi et al- are listed in Table 1 together with corresponding notations used in RMP.A/A(max)- A -- 5 -TableList of Symbols Used in RMP and in Salardi et aiRMP (also in this report) Sa 1 ard i e t a 1A Lk2 KTR1-6X, Y(2tt/a) / 1  - Aa (27!/A ) /l + Aa /(l - Aa)/ (1 + Aa)(1 - a2)/[A (1 + a2)]X' , Y 1 (L/AtOx1 , (L/Att)X 'z 96RR-0IU|> 0)1 z<j> 0)2 z0)1CO2a-  6 -2. TRAJECTORIES AND THE FOUR-DIMENSIONAL ACCEPTANCEExact solutions for fixed transverse coordinates x, x', y, x and forSince the aperture is assumed to be circular, both coordinate systems should give the same acceptance. The rotating system is chosen such that the V axis is always in the focusing direction. It is thus obvious that the maximum possible excursion of a particle will be in the Y d i rect i o n .From the expressions given by R.M. Pearce , exact solutions for X and Y areX (z) = k} (~D cos ip(z) +C sin ip(z)) + (l/k;,)( B cos <}>(z) +A sin 4>(z))Note here that M q = X(z=0), etc. and z is the axial distance in unitsof F 6 RR H E  that x^ and y^ have the same dimension (length) as M P andv . Stable motion exists only for |a| < and a is positive when Y ois the focusing direction. By rewriting the expression (1) in thegrotating coordinates X, X', Y, Y 1 due to L. Teng are given in RMP.y (z) = A cos <J) (z) +B sin <J> (z) +C cos ip(z) +D sin ij;(z) (1)where kj = (1 - 4 a P  (z) = kj z/ 2  A = y + x ' /2a7 O OC = -x 1/2ao4>(z) = k2 z/ 2  ,B = k2y 1/2a »oD = -(x + y'/2a)/k1 . (2)o oformX(z) = [(A2+B2 ) 2 k2] s i n + [kj (C2+D2) 2] s i n(3)Y (z) = (A2+B2) 2 cos (<J>“<t>0 ) + (C2+D2) 2 cos (tjJ~'PC)) with (p^  = tan 1 (B/A) and = tan 1 (D/C),one sees immediately that max.|Y(z)| = max. I X(z) | since 6 1 and k2 - 1. The four-dimensional acceptance at z = 0, when the aperture is R, is determined by the condition(A2 + B2)^ + (C2 + D2)^ £ R. (A)By evaluating the four-dimensional volume in x , x 1, y , y 1 spaceo o o obounded by the condition (A), Salardi et a 1.^ obtained the analytic expression for the acceptance A(x, x 1, y, y 1),A (x , x 1, y, y ‘) = (R‘V a 2) (2tt‘V 3 )  (Aa) 2 (1 - Aa ) h/ (1+Aa) \  (5)However, this is strictly true only for an infinitely long (that is, channel length >> A/kj) channel. For a channel of a finite length,cos (4>—4>0 ) and cos(^“>A0 ) may not take the limiting value (+1 or - 1 )at the same position within the channel so that the quantity on the left side of (A) could be larger than R. For example, with (J> = 0and ipo = 7t , ma x . | Y (z) | = | A | + |c| at z = 2nn /kj (n=l, 2 , --- )for (k2/k2) = even integers but max. |Y(z)| < jA| + |c| for any finite value of z when (k2/k]) = odd integers. Corresponding values of a are, for example, 0 . 1 5  and 0 .2 , respectively.- 7 -Although the four-dimensional region defined by (4) is generally of a complicated shape, its two-dimensional projection can be expressed in a simple analytic form for some special cases. Some of these are discussed in the Appendix.ACCEPTANCE OF FINITE CHANNELS AND THE "EFFECTIVE" ACCEPTANCEWhen the length of the channel is finite, the condition (4) is toorestrictive in defining the acceptable phase space region Sinceexact solutions for X and Y are given as a linear function of x , x ' ,J o oy^, y^, it is convenient to use polygons to represent acceptable areasQin (x, x 1) or (y, y') phase space .Unless the channel is very short (channel length % A/kj), the largest excursion within the channel would always occur in Y direction so that the condition for the acceptable region isR - Y(z) - - R for all values of z.For a given pair of (yQ , y^) , the condition (6) defines two parallellines in (x , x 1) space. The true acceptance in (x , x') space is the o o o oenvelope of these parallel lines evaluated at all values of z within the channel. For practical purposes, however, it is sufficient to take discrete values of z and regard the resulting polygon as a good approxi mation of the true acceptance. From Eq. (1), parallel lines evaluated- 9 -u(z) xo + v (z) x^ t w(z, yo , y^) = R and -R (7)where u(z) = - sin i|j(z)/k1v(z) = (cos <p(z) -  cos i p ( z ) ) / (2a)w = yo cos 4) (z) + y^ (k2 sin <J>(z) - sin ip( z)/k 1 )/(2 a).The procedure is of course quite similar if one wants to find theacceptance in 9A E S y^) space for a given pair of (x , x 1).The condition (7) is invariant under the transformation< v  '-o' x  - *  (~ v  - v  x >so that the acceptance has to be studied only for y > 0 or x > 0o oFurthermore, unless the channel is very short (channel length £ A/k^), the transformation(V  V  y;) “  <-x0 . Y0 , -v 'c )(xo- xo- yD ' -* (xo' X -  ~ vwould keep the acceptance unchanged. This can be seen from Eqs. (2) and (3) where these transformations merely change the sign of <j> and ipo so that two oscillations, cos (<p - <j>Q ) and cos (ip - ip ) , are shifted in phase by at most T RRC Thus, the area to be studied in the four­dimensional phase space is x , x ' , y . v 1 - 0o o ’ o 1 o-  10 -When the channel is to be matched to quadruples or bending magnets, only that portion of the (xQ , x^) polygons which are common to all values of (y , y^) could be utilized effectively. In general, it is very difficult to shape a beam such that its distribution in 9M q , M NI space depends in a specific manner on (yQ , y^). As a measure of this effective acceptance, one can consider the overlapping portion of a polygon with the standard polygon (for yQ - yQ 0) This trated in Fig. 3 where shaded areas are considered to be unacceptableeven though they satisfy the condition (6).k . COMPUTER PROGRAMS: HELIX 1 AND HELIX 2*4.1 HELIX 1 (see pages 21-26)For given values of the channel length and the parameter a, this program calculates acceptance polygons and their areas in (yQ , yQ ) space for mesh points in 9M q , M NI space. It can also be used for obtaining polygons in 9M q , M NI space for mesh points in (yQ , yQ ) space.Input parameters:card 1 (215, 3 F 1 0 .)IXY = 1 for polygons in (yQ > OP I space,2 for polygons in (x , x^) space.NZ(- 500) = total number of points along z direction wheretwo parallel lines (7 ) are to be evaluated.02 = interval of equally spaced points along z.i n meters- 1 1 -- 12 -Parallel lines are calculated atz = DZ, 2(DZ), 3 CDZ) , 3(DZ), . . . . , (NZ) (DZ) and the channel length (in units of A/rrr) is (NZ) CDZ) .SA = parameter a (between 0 and 0.25).R = aperture radius in meters (= 0 .1 ).The value of R is not really needed since M q , M NL yQ , y^ are all proportional to R, due to the linear approximation for the equations of motion. The choice of 0.1 is purely for convenience.card 2 (215, 2 F 1 0 .)NYNET (- 50) = total number of mesh points in M P direction(|XY = 1) or yQ direction (IXY = 2) including both end points.NYPNET (-50) = total number of mesh points in x^ direction(IXY = 1) or y'o direction (IXY = 2) including both end points.YMAX = largest value of M q (IXY = 1) or yQ (lXY = 2)of the mesh; usually this is equal to R.YPMAX = largest value of x^(lXY = l) or y^(lXY = 2)of the mesh; with R = 0.1, YPMAX = 0.03 " 0.05.Total number of points in the mesh is (NYNET) (NYPNET) with x = 0, YMAX/(NYNET-1) , 2 (YMAX)/ (NYNET- 1) , . . . . , YMAX; x' = 0, YPMAX/(NYPNET-1) , 2 (YPMAX)/ (NYPNET-1), . . , YPMAXofor IXY = 1Output parameters:For each mesh point (xq » x^) or (yQ , y^), coordinates (yQ , yQ) or (x , x^) of the acceptance polygon are given as well as its area. The maximum number of vertices is internally limited to 25 in the program.- 13 -IXY = 1 X n ~< o xp =Y = xo ’ YP = x 1 oIXY = 2 X = x ,o XP = x'o~< II < 0 YP = y;axial distance is in units of X/bv so that(dxQ /ds) = (k-n/X) x'o(dy /ds) = (bi r/X) y'O 7 owhere s = 96-NRRIU is the real distance (in meters, for example).^ 2 HELIX 2 (see paqes 27~35)This is identical to HELIX 1 but polygons at mesh points are overlapping portions only. The effective four-dimensional phase space acceptance is then numerically integrated. If the aperture radius R is given in meters, the four-dimensional acceptance in x 'o’ y0 ’ y;> sPace is in (meter)1*. In order to get the acceptance in (xq , dx^ /ds, yQ , dy^ /ds) space, one must multiply this by (Att/a)2 . Input cards are identical to those for HELIX 1 except that NYNET must be an odd number.Both programs could be used for a number of different parameters by just adding two input cards for each case. There should be a blank card at the end of all input data cards.-  ] k  -5. RESULTSSo far, two programs have been used for the following choice of parameters:R = 0.1, a = 0.05, 0.1, 0.15, 0.2, 0.215, 0.23,NZ = 314, DZ = 0.1, IXY = 2,YMAX = 0 . 1 ,  YPMAX = 0.03 or 0.04 depending on the choice of a,NYNET = 1 1 ,  NYPNET = 11 or 13- The channel is "long" for a = 0.05 " 0.2 but "short" for a = 0.23. Results for a = 0.215 and 0.23 may not be as accurate as for smaller values of a. Total computer time with Yale IBM 709MDCS) was about 80 minutes.All results are available in four IBM printouts. Normalized four­dimensional acceptances are shown in Fig. b as  a function of a; the solid line is for an infinitely long channel, the dashed line is for a channel of length 31 . *♦ A/4tt and the dotted line is the effective acceptance of the same finite channel. Although the finite channel gives a larger acceptance (dashed line) compared to an infinitely longchannel (solid line), its effective acceptance (dotted line) could beas small as one-half of that.It is interesting to compare the effective acceptance of a helical channel with the acceptance of a conventional AG channel. With a reasonable choice of the AG channel parameters,S/L = 0.05 S = distance between two adjacent quadrupolesL = length of each quadrupole R/L = \ R = radius of quadrupole aperture,- 15 -OZ' si • 01 ■- 16 -together with the assumption that two channels have the same period A of the pole rotation,A = 2X (magnetic period of the AG channel)= 6 L ’ (8 ) one gets the relationR/A = 1/24.This value is consistent with the assumption that the axial magnetic field Bz of the helical quadrupole is negligible. The maximum four­dimensional acceptance for each channel is " helical channel 10 .4 R V a 2AG channel 16.2 R V a 2The required field gradient of the helical channel is about 70% of what is needed for AG channel quadruples. On the other hand, only 70% of the total length will be occupied by quadrupoles in the AG channel. When the channel is to be used as a muon channel, it is desirable to have the same acceptance for pions and muons. For example, assume that p^/p^ = 1 . 3 3 (collection of forward-decaying muons). The acceptance one can get is then helical channel 9.3 r V a 2AG channel 15-3 R V a 2The AG channel is assumed to be infinitely long here. This is a good approximation when the channel has more than 8 magnets (four magnetic periods) and it is operated near the optimum momentum.It is of course possible to increase the effective acceptance of the helical channel by taking a smaller value for A than given by (8). The ratio R/A is then increased and the effect of the axial magnetic field may not be entirely ignored. Although the axial field seems to increase the focussing action of a helical channel , 1 the coupling of two directions x and y becomes more complicated (equations •of motion contain x 1 and y') and it is by no means obvious that this would cause an increase in the effective acceptance. Another dis­advantage of the helical channel is its inferior performance for p < p (optimum). The acceptance is zero at p = 0.75 p (optimum) whereas the AG channel still maintains S I X of the maximum acceptance at this momentum. The low momentum cut-off of the AG channel is p (cut-off) = 0 .6A d (optimum).The author is grateful to members of the TRIUMF Project at the University of Victoria for stimulating discussions.- 18 -REFERENCES1. R.M. Pearce, submitted to Nuclear Inst, and Methods, 1969.This will be referred to as RMP.2. Roland Cobb, private communication.3- For example, see Stopped Muon Channel", Los Alamos Design Status Report. V.W. Hughes, S. Ohnuma, K. Tanabe, H. Vogel (1969).4. G. Salardi, E. Zanazzi, and F. Uccelli, Nuclear Inst, and Methods, 59 (1968) , 152.5. See, for example, Internal Report Y-12, Yale University,October 1964, p. IV-182.6 . R. Helm, SLAC-4 ("Red" Stanford Report), August 1962.7- S. van der Meer, CERN 62-16, April 1962;E. Regenstreif, CERN 64-41, September 1964.8 . Lee Teng, ANAL-55, February 1959-9- A. Citron, J. Fronteau and J. Hornsby, CERN 63-3 0 , August 1963;J. Fronteau and J. Hornsby, CERN 62-3 6 , November 1962.- 19 -APPENDIXThe acceptable region in the four-dimensional space given by the condition (A) takes a complicated shape in general. However, its two-dimensional projection for some special cases can be represented by simple analytic expressions.1. Parallel Beams (x1 = y 1 = 0)---------------------o ' oly0 l + lx0 l/kl ~ R Fig. 5a2. Point Source (x = y = 0 )------------------ o 'ou = x V  (2a) , v = y V  (2a)(u2 + k^v2)^ + (u2 + v2/k2)^ ^  R Fig. 5bSince kjk2 = (1 - 16a2) 2 ^  1 , this area 1 ies between two el 1 ipsesu2 + k2v2 -  (R/2) 2u^ + v 2/k2 - (R/2) 2 broken lines in Fig. 5b3 - y^ = 0 u = xo/kj , v = x^/(2a)v ~ (R-yJ/2-!gu2/(R-y ) for v - -y o o ov - %u2/(R+yQ ) - (R+yQ )/2 for v -  -y . Fig. 5cNote that, for yQ = R, the acceptable area in (xq , x^) space reduces to a 1 i nex = 0 , -2aR ^ x 1 ^ 0 . o o-(R+y )/2- 21 -M C K I E R N A N  p Q A D G G b T y hoH K L I X l  - O^N . SO UR CE S T A T E M E N T  - IFN(S) - 0 9 / 0 3 / 6 9i £ « m N “ : x u » o u ; S i ? ? S i ; E1, 5 0 0 ,*M 's c r , -r 's o ''v p ' 5 ® * * * * ' * * . « > ' * < A , f1GC0 FORMAT! 2 I b , 4 F U  . 6 )1CC1 FO R M A T  ( 1H1 , 5X , U A = E 10 . 5 , 5X , 3HR = F 1 0 . b / / / )1002 F O R M A T ! / / l ! J X , l C H * * * * * * * * * * f3 X , 3 H Y  = F 1 0 . t>, 10 X , 4 H YP = F 1 0  5 3 X1 10H + * * * * * * * * * / ) riu,!>' 5 X *1003 F O R M A T (5 X ,1 O F A C C E P T A N C E ,F 2 0 . 5 )rGGy kunm6aV Di1u F -9DiTsCtgII1005 F O R M A T ! 5 H  ) XP / (bE ^  . »i )1006 FORMAT! / 5X , 1 '• F M A X . . ! s ■ A \ C F. = F 10 . 3 )1007 F O R M A T ( 2 I 5 * AT 1 C . 5 )1CC8 F O R M A T ( 5 X , 1 b h (NC AC', f P 1 A N C F )/)N X I =4500 R E A J (5, 1000 ) I XV ,N Z ,0 Z ,SA , R I F { I X Y . E w . O )STfF W R I T E (6 , 1001 )SA ,R W X I ( 1 ) = R fcX 1(2 ) = R WXI ( 3 ) = - R W X I ( 4 )=-R R X P I ( l)=0.b W X P 1 (2)=-0 . 5 W X P I (3 )=-0 . 5 W X P I {4 ) =c. 5 A 1 =SQRT ( 1. -4 . * S )A 2 = SQ R T (1 . +4.*SA )DO 10 K = 1 , N Z Z = F L U A T (K )*0/3C0 P S = A l* // 2.P b = A 2 * Z / 2 .C P S = C 0 S (P S )S P S = S I N ( P S )C PH = C O S (P H )S P H = S I N ( P H )C (K ) = - S P S / A 1 C ( K ) = ( C P H - C P S  ) / <r . / S A G 1 ( K ) = C P F.G2<K) = ( A 2 * S P H - S T S / a l  1/2./5A 10 C O N T I N U ER E A D (5 * 100 7 1 N Y N E T ,NY P N £ 1 , Y M A x ,Yp MA X CO 20 K = l , N Y N E T  20 YIK} = F L O A T ( K - l  )< Y M A X / F L O A T ! N Y  jtT-1 )CO 70 K = l , N Y P N E T  70 Y P (K )= FLCA T (K - ] } * Y F V A X / F L 0 A T (N Y P NI T- 1 )CC 3 0 1 = 1 , N Y N E T  YNOw = Y ( I )CC 40 J = 1 ,NY PNC 1 YFNCW = Y P {J )w R I T E (6 , 1 0 0 2 ) Y N C W ,YPNUW NX =NX IDO 51 L = 1,NX I w X (L )= W X I (L )WXP(L) = WXPI (I. )51 C O N T I N U ECC 50 K = 1 , NZ- 22 -V C K I E F N A N  PHY 5 0 0 - 2 4 8 9  DATE 0 9 / 0 3 / 6 9H E L I X  I - t: E h SOURCE STATE,-: NT - £ F N ( S »A Y (4 ) =PI F (I X Y , £ Q . I ) G J  TC 5 "A Y ( I ) =C ( K )A Y ( 2 ) = 0< K )AY ( 3 ) = Y I ( K ) Y F 12 ( K )GO TC 96 55 A Y ( I ) = G I ( K )A Y (2) =0 2 (K )A Y ( 3 ) = YNCrt*C l .<• ) • Y P ’ J W *0 ( K )fY s600 s l a e 0 v o c ,tr,/< ,W*P,AY )I F I N X . G T . 2 5 )  C/'i.L ' E D U C E ( N X , w/ , w X P , 2 5 >CALL A R E A ( \ X , w * , s X P , AC CE PI )IF ( A C C F P T . L E . C . i l IC 90 50 C O N T I N U E60 WRITE (6,100 3) Ao- i: IWR I Tfc (6 , 1 304 ) U  .* f K ) , K = 1 f NX )WRITF (6,1005) (r ■*•{ K) ,K=1 , NX)GC TC 40 90 WRITE (6 , 1008)40 C C N I I N U E  30 C O N T I N U E  GC TO 500 ENDC- 23 -P C K I E R N A N  P H Y 5 0 C - 2 4 8 9LACUT - r'FN SCUFCE S T A T E M E N T  - IF N {S iS L B R C U T I N E  C L T P I (M , A , Y ,A )D I M E N S I O N  X { I PC ) , Y ( I TO ) , A ( 4 ) , V ( 1 00 ) ,w < 1 0 0 ) 2 X IF ( P ) = - ( A 2 * C + ( A 3 - P } <= iJ X ) / D t T Y Y l F ( P ) = ( A i * C - < A 3 - P ) * 0 Y ) / D E T  A I = A ( 1 )A 2 = A ( 2 )A 2 = A ( 3 )N = CX (M+I ) = X ( 1 )Y ( y ♦ 1 J = Y ( 1 ) y p B p ♦ iCC 50 I=2tMM CX = X { I >- X ( 1 - 1  )CY = Y I I )-Y ( I-I )C = X( I Y ( I — I )— V < I )*X{ 1-1)E L I = A ( 1 ) * X ( I ) + {■ ( 2 ) * Y ( I ) + A ( 3 )E LIP= A ( 1 )*X ( 1-] >♦A ( 2 ) *Y ( I - 1 ) +A( 3 ) C E T = E L I - F L I P  I F ( A ( 4 } )5,55,5 5 J J = 0I F (E LI -A (4 ) 110, 1C,1210 J J= IIF(ELI + A(4) ) 12, 1 1 , I I11 JJ = 212 KK = 0I F ( E L I P - A ( 4 ) ) 1 9 , 1 5 , 2 015 K K = 1I F ( EL IP ♦ A ( 4 ) }2 J , 16, 1616 K K = 220 K = 3 * J J + K K + 1DO TC (50,2 3 , 2 4 , 2 5 , 5 0 , 2 6 , 3 0 ,  31 ,49) ,K23 N = N + 1V ( N ) = Z X I F (-A (4) ) fc < N ) = Y Y I F (- A (4 > )24 N =N+ 1 V ( N ) = 2 X I F I A I 4 )  ) t ( M  =YY If- ( A ( 4 ) )CC TT 5025 N=N+1V(N) = ZX IF (A (4 ) I X ( N ) = Y Y I F ( A ( 4 )  )26 N = N + 1 V ( N ) = Z X I F ( - A ( 4  ) ) k (N )= Y Y I F ( - A (4 ) )CC TC 5030 N = N + I V C N ) = Z X I F ( A ( 4 )  )M N ) = Y Y I F (  A { 4 I )CC TC 4931 N = N * 1V ( N ) = Z X I F ( — A ( 4 ) )M M = Y Y I F ( - A ( 4 )  )GC TO 495S .1.1 = 00 9 / 0 3 / 6 9- 2k -M C K I E P N A N  P H Y 5 0 0 - 2 4 8 9  D A TE 0 9 / 0 3 / 6 9LACUT - i.hiN SOURCE S T A T E M E N T  - IFN(S)IF(EL I )5 7, 56, 5656 JJ = 157 KK=0 I F ( E L I P ) 5 9 , 5 3 , 5 858 K K = 159 K = 2 * J J  + KK<-1GO TO (5 0 , 2 4 , 3C ,49 ) ,K49 N =N+ 1Y C N > = X ( I I * ( N > = Y ( I )50 C O N T I N U E  F = NI F ( N >7 5, 73 ,6 060 X M I N = l . E 1 0  Y M I N = 1 . E I 0  X M A X = - 1 . E 1 0  Y M A X = - 1 . E 1 0  CO 681= ItNI F (V ( T )— XMI N ) 6 1 , 6  2,6261 XM I N = V ( I )62 I F (V ( I ) - X M A  X )6 A , 6 4 , 6 363 X M A X = V ( I )64 IF(w( I ) - Y M I N ) 6 5  ,6 6 , 6 665 Y MIN = w ( I )66 I F ( kv ( I ) - Y M A X ) 6 E ,66,6767 Y M A X  = a 6 I )68 C O N T I N U ES I Z E 1= X M A X - XM IN SIZE2 = Y M A X - Y M  IN S I Z = S I Z b l + S I Z E 2  IF (S I Z . E C . O . O )  N = 1 M = 1X ( 1 )=V( 1)Y { 1 ) = W ( 1 )IF(N.LE.l) GO Tf 75 C C 7 2 I = 2 , NIFIS1ZE L.Lfc.0.Q.ANL'.SIZE2.GT.0.0) GO T J 80 IF(ABS(VI I ) - X ( M ) ) / S l / E ! - . O O O C 5 ) 8 1 , 7 0 , 7 0  81 I F < S I Z E 2 . L E . 0 . C ) GC TO 72 80 I F ( A B S ( > a ( I ) - Y ( M  )/S I 7 E 2 - . 0 Q 0 C 5 ) 7 2 ,  70, 70 70 V = M + 1X ( M ) = V ( I )Y ( M ) = W ( I )72 C O N T I N U E  75 RETURNEND- 25 -C K I t t - i \ - » H Y  5 '0-2489 DATEL A Jt.JCE - G! J Si.v.-CE S T A T E M E N T  - iFN(S)S U R Rl J 1 i M  E LL t E ( N » X , Y , T-‘ AXP T )C I >v E 0 S I A  .<( n o  ) , Y( iC-T)I F { N . L t .'' A X ° 1 ) F E T L !< N 45 CI ST = 5 * 1 ( ( X ( N )- X( l) ) * * 2 + ( Y ( N ) - Y ( i) ) * * ? _ )IK =Nd B q t 1CC 20 J. - I » MC I STC = 1 -'1 ( (> ( I ♦ 1 ) - x ( I ) ) * * 2 + ( Y ( I + i ) - Y ( I ) ) * * 2 ) IF (01S1 . 1ST ) GO I J 20C I S T = 0 J ,10 I K = I 20 CC N TINU cIF ( IK.LT. n  hi 2 8X ( 1 ) - ( X ( 1 I ♦ \ ! M i ) / 2 .Y ( 1 ) = (Y ( I » * ( < \ ) 1 / 2 .GO TO 35 25 X(IK) = (X< !*)♦*< IK+1) 1/2.Y ( I < ) = ( Y ( I i\ i t v ( K +  I ) >/2.I F ( IK . fcQ . • ) 11: T C 3 5L L = I K + 1 CC 30 J = LI , 1 X ( J ) = X { J + I )3 3 Y (JI = Y (J + 1 )35 i\ = N-lI F (A . G T . M A X P T ) CC TC 45PE TOP NLNC0 9 / 0 3 / 6 9- 26 -ns. 1 w j q c q Pi Y -.'GO-?** -;■* rM -■> :URCE S I AT tME NTL A R E A E OAT - 'Jl/ / 0 3 / 6 SSLOP C M  T fv (• A R A ( N f a * Y » A ) Cl MENS I UN X (1 : ,Y (1C 0 )Z = 0{ X ( I - i ) - :< ( I ) ) ) / 2 .20 CC N I r  u E A = Zni a( Q xEND- 27 -NCK J E M ! A :'i .r H Y 5y C - 2 A 6 9 D A T E  0 9 / 0  3 / 6 911 L L I X ’ - FN SOUkcE 6 T A T EME N T - IFNI $ )C U V M  N C( 50C i ,L)l 6*' J »GJ (V  ’■ > ,02(300) , w x ( 10 G } , •< X P ( 10 0) , A Y ( 4 } ,IF t NX »YNliW »Y PtOu , I X Y UI MENS 1 C N Y { 3 J ) , YP ( i 0 ) , iv X I ( A ) » wXP 1(4), ACCL p f ( 500 ) f WXF ( 100 ) ,l w XP F(ICO )L I ME NS i ON A P 1 S ( 5 1) , AP I rj l b - } ,  5 1 ) , A I Y ( 50 )rsGG k u e m c a ( 2 I 5, Al IE . b )L 0 01  F O R M A T  ( 1 H 1 ,  5 a , 5 H A = F 1 C L 5 0 1 F L 1 OO R M A O T L ( H H H ,ICO? FUR M aT ( / / 1 ■:X , 10 m ***<=****#*, 3 X , 3 HY =(-10.5,1 X.aHYP = F 1 0 . 5 , 3 X ,  1 1 O H * **** *** *= / )IOC 3 FGkMAT ( 5X , 1 C^AC .. E P T A NO c / { 5E20 .5 ) )ICG A FGKMAT < 5110 X / ( 5l 2 :" . 5 ) )1C T (  FGKMAT 5 ( a T  F=m / l b t 2 u . r> ) )1006 FORMAT </5X, 1A H MA X . 01 S T A N C E = F 10.3)1C 07 FORMAT i 2 I 5, Hi 1C . 5 )1C CB FORMAT I 5X , 1 ShIN'I ACCEPTANCE)/)NX 1 =A( TO RE AG (5 0 O CO )I XV 0 NZ 0 D Z 0 5 A 0 *IF { 1 XY . E W • 0 ) 5 T i WK ITi ( 6 , 1C) 1 ) S A , :■A F  ( O)MF  5 ,  =R F    , M   X X i ( A ) = - RrtXPI (1 >=0.1w X P 1 ( 2 ) = - C . 1WXPI <3)=-C. 1WXP I (A ) =C . iA 1 — S C. K T ( 1 . — A . * S A )A2 = SCRT ( 1 . + A . * S A )DO 10 < = 1, N •Z = FLi.A ) ( k ) L /.ICO P S - A 1 * Z Z 2 .PH = A 2* / / 2 .C P S = CO .'»( PS )SPS=SIfJ{PS)s e p B ,l 'g 9enISPn = si r i 9 j Q IG ( K ) =- b P S / A 1D U )  =( G P t l - C P O / b  . / S AG 1 ( K >=C PHG 2 < K ) = ( A 2* ) Pi-oP ;/ A I ) / 2 . / S A iO CGiN I I o u rKb AC (5 * IOC/ ) U N !  1 , . YPNF T , Y <)AX , Y P M A X  DO 2 C K=l»NY\iCT 20 Y ( K ) =F L GAT ( K- I) * YM A x / M  0 A TINY f- 1 !on 7 0 K= 1 , -JYi’Nt: T70 Y P ( K ) = F L t)A f (r, - i ) * Y K • A X /FL S p T ! i YPN E T- 1 )YNGW =0 .YPN0w= V .NX = N XI)s D su 0 B rz oF 2 p F ­ 0 I B C F 2 9 0 I5 00 WXP ( L ) =W XP I ( u. )CO 6 00 K=1,NZ CALL PUL Y U  )- 28 -m s . 2 i / o 6 o  e v O D S u t 2439 DAT t 0 9 / 0 3 / 6 9H r L {X c - t r N SO UR C E S T A T E M E N T  - I F N ( S )CALL A ,• A ( j , c F e 96ss V6 00 CCNTI-jOF 6 01 N a F - * XDO fc L 2 L = 1» N X F rtXF(I ) = 4 X ( L ). 6 02 WXPF (L ) =WXP { L )APT S ( 1 ) =ACC wRITfc(6»2Ci. u 6 s ( C 20 uO FORMAT ( //5X, 2 uHC ENT .AL AC.LEPI A-NCt =012.5)to K I T E { 6 » I ) 4 ) < .*) a F ( L ) * L = i » N X F ) I O   t  O  m  , 5 !" #  $ A   %  , & %  M  " ' F A ,DO 3 0 I — I »'4 V M •- TOou 2q BO 9 2 I403 CO 4 C J = I » \i YP ME T YPi\iGto=YP( J)WRITE* b f ICO 2 ) YN a, YP NC kIF ( Y NO tv . I: Q . C . .4 j u . Y t ’ N U X . E W • C • ) OU Tu 4 0 N X = N XI A R A P = 1U ) 0 .DC 5 1 L = I » J X I W X ( L ) = w X I ( L ) toXP(L)=„XPI(L)51 C O N T I N U ECO 5 u K = I t N I  CALL P O L Y (K )CALL AR L A (N X » OX ? WX P *ACCE P T (K ) )I F (A L C c P T K ) .Lc .0. >00 TO 90 50 C C N T I I O F60 CALL 1 NTPO( NxF» .. XF « .. X P F , N X ? w X • w' X P )I F ( N X ( 0 " L c 5 ) L A L L R c D u C L ( i F  si F ) a X $  2 ( )CALL A i< L A ('( X » W X.? i, X P »AP I ) A  * $ = L % + L  .0)0 = ,  9 0APT S (J )= 4 PT J NOto - JtoRITC(6 , 2 0 0 1 )APT 2001 FORMAT ( 5 X , I 2H AC ,, cP T A NC F =L12 .5 )W jO 1 L ( 6 , 1 0 ■ 4 ) ( a a ( K ) • ft. = ] t \ A i  T   5 )L L  - -  ( .   H0  $ 5 m  "  / M   1 'F ,40 CLMl INOP 93 M  I T r. { (■ , ].C ; 8 )JLSI — J !::j  ^p i APT S ( J L .> T ) = o .J C K = J j OJ-2* ( J N J  ,/2 )IF ( J (X • L 0 . 1 ) J L S f = J 'V 'ti CO 4 8 J J J = I ,J LS f 4 80 AP I N T ( I » J J ) = A P T S ( J J )JTUT = J L S T - ,2 A IY ( i ) = 0 .CO 8 53 K K J = I , J T >T, 2453 A IY ( I) = A I Y( I ) + A P 1:4 I ( I , K K J > *■ 4. * A P I M  ( I »i\ K J + L ) + A ° I M T ( I » KX J ♦ 2 )A IY ( I ) = A I Y ( 1 ) * A n S ( Y P ( j ) — Y P ( I) )/_>.30 C O M  IN OF A I Y Y P = C  .I TO T -• J YNE T- 2 00 4 54 1 = 1? I I 0 T ? 24 54 A IY Y P= A I Y YP + A I YI I )+ 4 . * A I Y ( 1 t i ) f A I Y ( I +2 )- 29 -Y K J L fti. A inI : L I A-’: '? Y P •- <’ I Y M* -H” S ( 0 L /  =  L 5 ( 0 O   0   , *  1  O % O A % A  Hm2m 3 H H H H   4 F 0  ,A IV Y $ O ( % /  = 5 * . 5 5    *, k I 1 t ( t , 1 1 I 2 ) A I Y   O "  6 $ 5 * )  3 H H O- *  %  (      (  7  8 ' 9I H Y 5  ; C - ? 4 8 9  D A T Et i i .  '.> tJi" i .  r S T A T i  «L  T -  I F f M ( S ); ■:<:)). ■.I A ! 1 ’ i r. . 1 1 5 . ■.))0 9 / 0 3 / 6 9- 30 -NCK i tF.fi AN; PHY 5 3 0-<-49 9 D A T E  0 9 / 0 3 / c>9L A C U T - t E N S u i J K C E S T A T E M E N T  - I F N (S ) —SLBRCoT I NE C U TP ( M , X , Y ,A)L I MENS ION X ( U G ) ,Y ( I C C ) , A (A ) , V ( L O G ) , n (IGO) Z X I F ( P ) = - < A 4*0+1 A 3 - P )* D X )/DtI YY1 E (P ) = {A1*C-( A 3-P )*0Y ) /DE T A 1 = A (1 )A 2 = A (2)A 3 = A ( 3 )N = 0X ( M + 1) = X ( 1 )V C M+ 1 ) = Y ( I )V P = M +1CO 5 C 1 = 4 *> ■'CX = X <I }- X { i- i )L Y = Y (i )- Y ( I - i )C = X ( 1 )* Y ( I- I ) - Y ( I )* X ( I - 1) cL I=A( 1 ) * X< I ) + A ( 2 ) * y ( i ) + A ( 3 )LL I P =A ( I ) *X II -1 ) + A { ; ) *> i 1 - 1 ) + A ( 3 )DE T = EL I - L L I P I F ( A (A ) ) 5 , 5 6 , 5 5 J J = C1 F ( E L i - A < A ) ) i :•, i C , 12 10 JJ=1I E  ( t  L I  + A ( A ) ) I  A » 1 I T 1 i  LI J J = 2 12 K K = CI E ( t L I P - A ( A  ) J15, 15,^015 KK= 1IF I E 1.1 P + A ( A ) ) 2 C , i o , 1 o16 KK = 220 K = 3 * JJ +K K + LGO T L ( 5 0 * 2 3 * 2 A * /5 * 6 0 * 2 6 * 3 0  *31*49) * K 23 N = N+iV ( N ) =/. X I E I - A ( A ) ) fc ( N ) =Y Y I E ( - A ( A ) )2A N = N + 1V ( N ) - L  X F F ( A ( A ) ) r. ( u ) =Y Y I F (A ( A ) !GU TG 604 5 N = N + 1V ( \ ) = Z X I r (A ( A ) ) w ( N ) —Y Y 1 F ( A (<t ) )26 N = N ♦ 1V (N ) =7 X I E (- A ( A ) )X < N ) =i Y [ E ( - A < A ) )GO TO 6030 *\ =N + iV ( N ) =L X I E ( ;a ( A ) ) y. ( N ) =Y Y I f ( A ( A ) )GL !C A 931 i\ = N + 1V(N ) = Z X I F (- A 1 A) ) r. ( N ) =Y Y IE (-A( A) )CL Tl AO65 j J = CM C K U  ; .A iV‘ jQ-24 -■ -/S 'JRCh - 1 /■ F £ f ■* ,TDATfc 0 9 / 0 3 / 6I F( i. L . ) •> / t ■ .56 JJ=157 K K= CI F ( E L I P ) 5 9 , . ‘ . ,58 K K = 159 K = 2 * J j + K K + 1GC T f. £ lju 1 1  zi i _ > )  i49 N=N|+1~ 9 1p I tF 9 2 IW ( N ) = Y ( 1 )50 LCN7 i'>*Jf r = NIF-(M )75, 7 5,6060 XKIN = 1 . L. 10V MIN = 1 .L 10  X K A X =— I . fc 10V M A X =- I . b 10 DO 6 ei = I , NI F ( V ( I ) — X MI N ) o 1 * t 2 f ( 261 X M I N = V ( 1)t:2 1 F ( V ( I ) - X V A X ) 6 4 i, >: 36 3 X M A X =V ( I )64 1 F ( W { I )-YYi N ) 6 3 . , i o65 YM IN =*'(!)6 6 I F ( W ( I ) - Y 7 \ X ) j b , ' 8 , 6 7o 7 Y M A X =.; ( i )-.8 CCiMT !Nlit.IZE l=x 3 A X -  X,-. I S I Z £ 2 = Y M A X - Y M 1,-.. IL = 6 I Z t 1 + 9 I / 2 I F ( S I Z . t w . o . ) . - 1= 1A ( 1 ) =V ( i )V ( 1 ) =* ( 1 )I F ( N .L L . i ) ,CS i 7 3 DC 7 ^  I = ^  , Ni F ( S iZ F 1 . Lr . . I ZrZ. GT . v. 0) G u T ■. J 6 jIF ( ABS ( V ( I) - / (i-1 ) I / 3 i L t I - . Co ) 03 ) 6 1 , 70 » 781 I F ( S IZ E 2 . L t . 0 . C i G J I 0 7 280 IF ( Afco ( ,< ( I) -Y { M t ) / 3 I / t 2- . CO b) 72 , 7 0 ,  I  .)70 N = M+1X ( M J =V ( I )V ( M ) =W ( I )72 CON I IMUE7 5 !? E f U P 1ENL)- 32 -M C K I fc R "I A M PHY5 ,'0-2489 _ U A 7 E  0 9 / 0 3 / 6 9L A U U C E  - E E N SQ UR Ct S T A T E M E N T  - IF N ( S )SOHR CUT 11\ t R E C U C E < N » X » Y  » M A X P T )LU ME NS I i >\ a ( L . u ) , Y ( i GO )1 E ( N . L E . V A a P T  ) k E T U ^ N  46 C I S T = S W < T {( x( U-.<( 1) )**?+( Y(N)-Y( L)IK = N M =N- 1CO 2 C I - L » A-C I S T C= S jRT ( ( X ( 1 + i ) - a  ( I ) ) * * 2 + ( Y ( 14- L ) - Y ( I ) ) * * 2 ) I F I O I S T C . G O  .0 1 ST ) Go TO 2 0 CIST -iJ I j T C IK = I 20 C O N T I N U EI K  1 K . L T . M  G J  1C  2 5  X ( l)=l X( 1 ) + X ( N ) ) 1 2 .Y ( 1 > = ( V ( 1 ) + Y ( N ) ) / 2 .GO TC 3525 X( IK)— (XI IK ) + X { I K 1) ) / 2 •Y ( I K ) = ( Y ( IK ) + Y ( I K + 1 / ) / 2 . lE(IK.fcj.M) GO TO 3.>L L = I K + 1 DO 3 C J=LL, v X ( J ) =X ( J + 1 )30 Y { J ) =Y ( J ♦- 1)35 N=N-1IE ( N .oT .MAX P T ) • , TC 4 5'•’ETC RNEND- 33 -M C K I E R N A N  HriYl5 ..0-2489 DATC A - c A  - Li- N S O U N C t  S T A T E M E N T  - IFNI SIS L 8 * D J T  I N E  /*!<; A ( N , /., Y , A J CIMtKS IJN X ( I•_ 0 ) , r ( 1 .I C )Z = 0DO 2 C 1=2.NL = Z + AS S < (X ( 1 )- X ( i — 1) ) - ( Y ( I- 1 J- Y (I ) I — ( Y ( l ) - Y ( l - l )  I * ( X { f - 1 ) - X < i I I I / 2 .20 CCNTIiMUt A = ZR E T U R NEND0 9 / 0 3 / 6 9- 3^ -MCK ItHNAN PHY 50 0-243 ‘9 D A T E  0 9 / 0 3 / 6 9L A PPLY - t F N S U U K C t  S T A T E M E N T  - IFNIS)SUBR CUT I NE P G L Y ( MCOMMC'i C ( 50 C ) * 0 ( 600 ) *01 (SCO) » (600) * KX ( ICO ) *wXP( 100) * AY ( A ) ,IP t N X * Oo=CS; »Y PiN JW t IX Y A Y {A ) = »IF ( IXY . ;Q.l )GC r... 1C A Y ( i ) = C ( K IA Y ( 2 ) ~ D ( K }A Y ( 3 ) = Y N C W * G  L ( K ) Y ( K )Gu TC 20 10 AY ( i ) = G L ( K )A Y ( 2 ) - G 2 ( K )A Y ( 3 )= Y N C W * C  ( < ) + /P\Lrt <=0( K )20 C ALL C u T PC ( NX , wX f ip X (», A Y i2k 9oF C, 2 CDD I s 6 0 0  v ( u UC E ( NX , w X , k\X P , 2 5 )RETUP'iLND- 35 -■MCKIfcf MAN PHY5 iC-2489 D A T E  0 9 / 0 3 / 6 9I M P 6V - t F N SlJUkCt 3 T A T tMLNT _ 1FN(S) —SLBR LU 7 I N t I M 7 P L ( K 1 , X , Y , N 2 , V , A )D I MLi\S I ■. jN X ( 1 ' 3 ) »Y { 1 0 » , V ( 100 ) , X ( 100 ) » A ( 4 IJ = M  +1 X ( J I — X ( 1 )Y ( J J =Y ( 1 JCG 2 0 I  = n r JA ( 1 ) = Y {• I ) -Y ( I - 1 J* 5  , M F 5 O  O  ,  F 5  O * 5 1 , M 5 5    O , # F 5 O ,  F 5 O  O ) # 5 5  ,A ( 4)=0 .CALL GUI P C I M - > V ,X ,A )1 F ( N < 1 2 1 , 2 1 , * :  020 CON 7 1NUL21 H L T U f- j END

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