Euler Coordinates The Euler coordinate system is introduced here as an orthogonal coordinate system in which the Euler line is the x-axis, the orthic axis is the y-axis, and the origin is X(468). The positive direction on the x-axis is that of the vector X(468)-to-X(2) ; the positive direction of y-axis is that of the vector X(468)-to-X(16272) when (b-c)(c-a)(a-b) is positive, otherwise that of the vector X(468)-to-X(16309). If P = (u : v : w) (barycentric coordination), then the Euler coordinates (xe, ye), of P, are defined by xe = S A u + S B v + S C w ye = -S A (S B -S C )u - S B (S C -S A )v - S C (S A -S B )w. Here xe = kd, where k = (E-8F) 1/2 = 6∣X(2)X(3)∣ and d is the signed distance from P to the orthic axis. Likewise, ye = kSd’, where d’ is the distance from the Euler line. If P is a triangle center, X, then there exist symmetric function G(a,b,c) and H(a,b,c) and nonsymmetric function K(a,b,c) such that X = G(a, b,c)S 2 +H(a,b,c)+K(a,b,c), Let N = (3G+H)S 2 +$K$, the Euler coordinates of X are then given by xe = [(EG+FG+3FH)S 2 +$K$]/N ye = - [$S A (S B -S C )K$]/N In particular, for X on the Euler line, K(a,b,c) = 0 and (G, H) are the Shinagawa coefficients for X. If X is not on the Euler line, then X is classed according to the denominator N shown just above. If X is indexed in ETC as X(n), then the Euler coordinates of X are written as ( xe(n), ye(n) ). As an example of a class, Table 1 Euler coordinates (xe(n), 0) of X(n) in N=S 2 -class, on the Euler line, Shinagawa coefficients (G(n), H(n)), and GXe(n) =∣xe(2)xe(n)∣/(E-8F) X(n) xe(n) (G(n), H(n))S 2 GXe(n) X(2) (E+F)/3 (1, 0)/3 0
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Euler Coordinates
The Euler coordinate system is introduced here as an orthogonal coordinate system in
which the Euler line is the x-axis, the orthic axis is the y-axis, and the origin is X(468).
The positive direction on the x-axis is that of the vector X(468)-to-X(2) ; the positive
direction of y-axis is that of the vector X(468)-to-X(16272) when (b-c)(c-a)(a-b) is positive,
otherwise that of the vector X(468)-to-X(16309).
If P = (u : v : w) (barycentric coordination), then the Euler coordinates (xe, ye), of P,
are defined by
xe = SAu + SBv + SCw
ye = -SA(SB-SC)u - SB(SC-SA)v - SC(SA-SB)w.
Here xe = kd, where k = (E-8F)1/2
= 6∣X(2)X(3)∣ and d is the signed distance from P
to the orthic axis. Likewise, ye = kSd’, where d’ is the distance from the Euler line.
If P is a triangle center, X, then there exist symmetric function G(a,b,c) and H(a,b,c)
and nonsymmetric function K(a,b,c) such that
X = G(a, b,c)S2+H(a,b,c)+K(a,b,c),
Let N = (3G+H)S2+$K$, the Euler coordinates of X are then given by
xe = [(EG+FG+3FH)S2+$K$]/N
ye = - [$SA(SB-SC)K$]/N
In particular, for X on the Euler line, K(a,b,c) = 0 and (G, H) are the Shinagawa
coefficients for X. If X is not on the Euler line, then X is classed according to the
denominator N shown just above.
If X is indexed in ETC as X(n), then the Euler coordinates of X are written as ( xe(n),
ye(n) ). As an example of a class,
Table 1 Euler coordinates (xe(n), 0) of X(n) in N=S2-class, on the Euler line,
Shinagawa coefficients (G(n), H(n)), and GXe(n) =∣xe(2)xe(n)∣/(E-8F)
X(n) xe(n) (G(n), H(n))S2 GXe(n)
X(2) (E+F)/3 (1, 0)/3 0
X(3) (E-2F)/2 (1, -1)/2 1/6
X(4) 3F (0, 1) - 1/3
X(5) (E+4F)/4 (1, 1)/4 - 1/12
X(20) (E-5F) (1, -2) 2/3
X(30) ∞ (1, -3) ----
X(140) 3E/8 (3, -1)/8 1/24
X(376) (2E-7F)/3 (2, -3)/3 1/3
X(381) (E+10F)/6 (1, 3)/6 - 1/6
X(382) (-E+14F)/2 (-1, 5)/2 - 5/6
X(546) (E+16F)/8 (1, 5)/8 - 5/24
X(547) (7E+16F)/24 (7, 3)/24 - 1/24
X(548) (5E-16F)/8 (5, -7)/8 7/24
X(549) (5E-4F)/12 (5, -3)/12 1/12
X(550) 3(E-4F)/4 (3, -5)/4 5/12
X(631) (2E-F)/5 (2, -1)/5 1/15
X(632) (7E+4F)/20 (7, -1)/20 1/60
X(1656) 3(E+2F)/10 (3, 1)/10 - 1/30
X(1657) 3(E-6F)/2 (3, -7)/2 7/6
X(3090) (2E+5F)/7 (2, 1)/7 - 1/21
X(3091) (E+7F)/5 (1, 2)/5 - 2/15
X(3146) (-E+11F) (-1, 4) - 4/3
X(3522) (3E-9F)/5 (3, -4)/5 4/15
X(3523) 3(E-F)/7 (3, -2)/7 2/21
X(3524) (4E-5F)/9 (4, -3)/9 1/9
X(3525) (4E+3F)/11 (4, -1)/11 1/33
X(3526) (5E+2F)/14 (5, -1)/14 1/42
X(3528) (4E-11F)/7 (4, -5)/7 5/21
X(3529) (2E-13F) (2, -5) 5/3
X(3530) (7E-8F)/16 (7, -5)/16 5/48
X(3533) (6E+3F)/17 (6, -1)/17 1/51
X(3534) (5E-22F)/6 (5, -9)/6 1/2
X(3543) (-E+17F)/3 (-1, 6)/3 - 2/3
X(3544) (4E+19F)/17 (4, 5)/17 - 5/51
X(3545) (2E+11F)/9 (2, 3)/9 - 1/9
X(3627) (-E+20F)/4 (-1, 7)/4 - 7/12
X(3628) (5E+8F)/16 (5, 1)/16 - 1/48
X(3830) (-E+26F)/6 (-1, 9)/6 - 1/2
X(3832) (E+13F)/7 (1, 4)/7 - 4/21
X(3839) (E+19F)/9 (1, 6)/9 - 2/9
X(3843) (E+22F)/10 (1, 7)/10 - 7/30
X(3845) (E+28F)/12 (1, 9)/12 - 1/4
X(3850) 3(E+8F)/16 (3, 7)/16 - 5/48
X(3851) (3E+18F)/14 (3, 5)/14 - 5/42
X(3853) (-E+32F)/8 (-1, 11)/8 -11/24
X(3854) 3(E+9F)/17 (3, 8)/17 - 8/51
X(3855) (2E+17F)/11 (2, 5)/11 - 5/33
X(3856) (5E+56F)/32 (5, 17)/32 - 17/96
X(3857) (5E+44F)/28 (5, 13)/28 - 13/84
X(3858) 3(E+12F)/20 (3, 11)/20 - 11/60
X(3859) (7E+64F)/40 (7, 19)/40 - 19/120
X(3860) (7E+88F)/48 (7, 27)/48 - 9/48
X(3861) (E+40F)/16 (1, 13)/16 - 13/48
X(5054) (7E-2F)/18 (7, -3)/18 1/18
X(5055) (5E+14F)/18 (5, 3)/18 - 1/18
X(5056) 3(E+3F)/11 (3, 2)/11 - 2/33
X(5059) 3(E-7F) (3, -8) 8/3
X(5066) (5E+32F)/24 (5, 9)/24 - 1/8
X(5067) (4E+7F)/13 (4, 1)/13 - 1/39
X(5068) 3(E+5F)/13 (3, 4)/13 - 4/39
X(5070) (7E+10F)/22 (7, 1)/22 - 1/66
X(5071) (4E+13F)/15 (4, 3)/15 - 1/15
X(5072) (5E+26F)/22 (5, 7)/22 - 7/66
X(5073) 3(-E+10F)/2 (-3, 11)/2 - 11/6
X(5076) (-E+38F)/10 (-1, 13)/10 - 13/30
X(5079) (7E+22F)/26 (7, 5)/26 - 5/78
X(7486) (5E+11)/17 (5, 2)/17 - 2/51
X(8703) (7E-20F)/12 (7, -9)/12 1/4
X(10109) (13E+40F)/48 (13, 9)/48 - 1/16
X(10124) (17E+8F)/48 (17, -3)/48 1/48
X(10299) (6E-9F)/13 (6, -5)/13 5/39
X(10303) (5E-F)/13 (5, -2)/13 2/39
X(10304) (5E-13F)/9 (5 -6)/9 2/9
X(11001) (4E-23F)/3 (4, -9)/3 1
X(11539) (13E+4F)/36 (13, -3)/36 1/36
X(11540) (35E+8F)/96 (35, -9)/96 1/32
X(11541) (-4E+35F) (-4, 13) - 13/3
X(11737) (11E+56F)/48 (11, 15)/48 - 5/48
X(11812) (19E-8F)/48 (19, -9)/48 1/16
X(12100) (11E-16F)/24 (11, -9)/24 1/8
X(12101) (-E+80F)/24 (-1, 27)/24 - 3/8
X(12102) (-E+56F)/16 (-1, 19)/16 - 19/48
X(12103) (7E-32F)/8 (7, -13)/8 13/24
X(12108) (13E-8F)/32 (13, -7)/32 7/96
X(12811) (7E+40F)/32 (7, 11)/32 - 11/96
X(12812) (11E+32F)/40 (11, 7)/40 - 7/120
X(14093) (17E-46F)/30 (17, -21)/30 7/30
X(14269) (E+46F)/18 (1, 15)/18 - 5/18
X(14869) (11E-4F)/28 (11, -5)/28 5/84
X(14890) (55E-8F)/144 (55, -21)/144 7/144
X(14891) (23E-40F)/48 (23, -21)/48 7/48
X(14892) (17E+80F)/72 (17, 21)/72 - 7/72
X(14893) (E+64F)/24 (1, 21)/24 - 7/24
X(15022) (5E+17F)/19 (5, 4)/19 4/57
X(15640) (-5E+49F)/3 (-5, 18)/3 - 2
X(15681) (7E-38F)/6 (7, -15)/6 5/6
X(15682) (-2E+25F)/3 (-2, 9)/3 - 1
X(15683) (5E-31F)/3 (5, -12)/3 4/3
X(15684) (-5E+58F)/6 (-5, 21)/6 - 7/6
X(15685) (11E-70F)/6 (11, -27)/6 3/2
X(15686) (11E-52F)/12 (11, -21)/12 7/12
X(15687) (-E+44F)/12 (-1, 15)/12 - 5/12
X(15688) (11E-34F)/18 (11, -15)/18 5/18
X(15689) (13E-50F)/18 (13, -21)/18 7/18
X(15690) (17E-64F)/24 (17, -27)/24 3/8
X(15691) (19E-80F)/24 (19, -33)/24 11/24
X(15692) (7E-11F)/15 (7, -6)/15 2/15
X(15693) (13E-14F)/30 (13, -9)/30 1/10
X(15694) (11E+2F)/30 (11, -3)/30 1/30
X(15695) (19E-62F)/30 (19, -27)/30 3/10
X(15696) (7E-26F)/10 (7, -11)/10 11/30
X(15697) (11E-43F)/15 (11, -18)/15 2/5
X(15698) (10E-17F)/21 (10, -9)/21 1/7
X(15699) (11E+20F)/36 (11, 3)/36 - 1/36
X(15700) (19E-26F)/42 (19, -15)/42 5/36
X(15701) (17E-10F)/42 (17, -9)/42 1/14
X(15702) (8E-F)/21 (8, -3)/21 1/21
X(15703) (13E+22F)/42 (13, 3)/42 - 1/42
X(15704) (5E-28F)/4 (5, -11)/4 11/12
X(15705) (13E-23F)/27 (13, -12)/27 4/27
X(15706) (25E-38F)/54 (25, -21)/54 7/54
X(15707) (23E-22F)/54 (23, -15)/54 5/54
X(15708) (11E-7F)/27 (11, -6)/27 2/27
X(15709) (10E+F)/27 (10, -3)/27 1/27
X(15710) (14E-31F)/27 (14, -15)/27 5/27
X(15711) (29E-52F)/60 (29, -27)/60 3/20
X(15712) (9E-12F)/20 (9, -7)/20 7/60
X(15713) (23E-4F)/60 (23, -9)/60 1/20
X(15714) (31E-68F)/60 (31, -33)/60 11/60
X(15715) (16E-29F)/33 (16, -15)/33 5/33
X(15716) (31E-50F)/66 (31, -27)/66 3/22
X(15717) (5E-7F)/11 (5, -4)/11 4/33
X(15718) (29E-34F)/66 (29, -21)/66 7/66
X(15719) (14E-13F)/33 (14, -9)/33 1/11
X(15720) 3(3E-2F)/22 (9, -5)/22 5/66
X(15721) (13E-5F)/33 (13, -6)/33 2/33
X(15722) (43E-38F)/102 (43, -27)/102 1/34
X(15723) (23E+14F)/66 (23, -3)/66 1/66
X(15759) (25E-56F)/48 (25, -27)/48 3/16
X(16239) (11E+8F)/32 (11, -1)/32 1/96
Table 2 Euler coordinates (xe(n), ye(n)) of X(n) in N={(E+F)+(3)1/2
S}- class
X(n) xe(n)*N ye(n)*N/(SB-SC)(SC-SA)(SA-SB)
X(13) [{(3)1/2
(E+4F)S}+6S2]/6 -1/2
X(15) [{(3)1/2
(E-2F)S}+6S2]/2 -1/2
X(298) [(E+F){3(E+F)+(3)1/2
S}-6S2]/3 1
X(396) [(E+F){3(E+F)+2(3)1/2
S}+6S2]/6 -1/2
X(616) [{3(E+F)2+(3)
1/2(2E-7F)S}-6S
2]/3 1
X(618) [{6(E+F)2+(3)
1/2(5E-4F)S}- 6S
2]/12 -1/2
X(621) {(E+F)2+2(3)
1/2FS}-2S
2 1
X(623) [{(E+F)2+(3)
1/2(E+4F)S} -2S
2]/4 1/4
X(3180) [-(E+F){3(E+F)-(3)1/2
(2E-7F)S}+12S2]/3 -2
X(5318) 3(3)1/2
FS+S2 -1/2
X(5459) [{2(E+F)2+3(3)
1/2(E+4F)S}+6S
2]/12 -1/4
X(5463) [{4(E+F)2+3(3)
1/2(E-2F)S}-6S
2]/6 -1/2
X(5473) [6(E+F)(E-2F)+(3)1/2
(5E-22F)S}-6S2]/6 1/2
X(5478) [{18(E+F)F+{(3)1/2
(E+28F)S}+6S2]/12 -1/4
X(5611) [(E-2F){-(E+F)+(3)1/2
S}+4S2]/2 -1
X(5617) [(E+F){3(E+2F)+2(3)1/2
S}-6S2]/6 1/2
X(6669) [{6(E+F)2+(3)
1/2(7E+16F)S}+6S
2]/24 -1/8
X(6671) [{2(E+F)2+(3)
1/2ES}+2S
2]/8 -1/8
X(6770) [(E+F){-9F+(3)1/2
S}+6S2]/3 -1
X(6771) [(E+F){3(2E-F)+4(3)1/2
S}+6S2]/12 -1/4
X(7684) [{6(E+F)F+(3)1/2
(E+4F)S}+2S2]/4 -1/4
X(11542) [{(3)1/2
(E+4F)S}+4S2]/4 -1/2
X(13103) [{9(E+F)F- (3)1/2
(E-26F)S}+6S2]/6 -1
X(13350) [{(E+F)(2E-F)+(3)1/2
(E-5F)S}+2S2]/4 1/4
X(14539) [{(E-2F){2(E+F)-(3)1/2
S}-2S2]/2 1/2
X(16001) [{-(E+F)(E-2F)+12(3)1/2
FS}+6S2]/4 -3/4
Table 3 Euler coordinates (xe(n), ye(n)) of X(n) in N={(E+F)-(3)1/2
S}- class
X(n) xe(n)*N ye(n)*N/(SB-SC)(SC-SA)(SA-SB)
X(14) [-{(3)1/2
(E+4F)S}+6S2]/6 -1/2
X(16) [{-(3)1/2
(E-2F)S}+6S2]/2 -1/2
X(299) [(E+F){3(E+F)-(3)1/2
S}-6S2]/3 1
X(395) [(E+F){3(E+F)-2(3)1/2
S}+6S2]/6 -1/2
X(617) [{3(E+F)2-(3)
1/2(2E-7F)S}-6S
2]/3 1
X(619) [{6(E+F)2-(3)
1/2(5E-4F)S}- 6S
2]/12 -1/2
X(622) {(E+F)2-2(3)
1/2FS}-2S
2 1
X(624) [{(E+F)2-(3)
1/2(E+4F)S} -2S
2]/4 1/4
X(3181) [-(E+F){3(E+F)+(3)1/2
(2E-7F)S}+12S2]/3 -2
X(5321) -3(3)1/2
FS+S2 -1/2
X(5460) [{2(E+F)2-3(3)
1/2(E+4F)S}+6S
2]/12 -1/4
X(5464) [{4(E+F)2-3(3)
1/2(E-2F)S}-6S
2]/6 -1/2
X(5474) [6(E+F)(E-2F)-(3)1/2
(5E-22F)S}-6S2]/6 1/2
X(5479) [{18(E+F)F-(3)1/2
(E+28F)S}+6S2]/12 -1/4
X(5615) [(E-2F){-(E+F)- (3)1/2
S}+4S2]/2 -1
X(5613) [(E+F){3(E+2F)-2(3)1/2
S}-6S2]/6 1/2
X(6670) [{6(E+F)2- (3)
1/2(7E+16F)S}+6S
2]/24 -1/8
X(6674) [{2(E+F)2- (3)
1/2ES}+2S
2]/8 -1/8
X(6773) [(E+F){-9F- (3)1/2
S}+6S2]/3 -1
X(6774) [(E+F){3(2E-F)- 4(3)1/2
S}+6S2]/12 -1/4
X(7685) [{6(E+F)F- (3)1/2
(E+4F)S}+2S2]/4 -1/4
X(11543) [{-(3)1/2
(E+4F)S}+4S2]/4 -1/2
X(13102) [{9(E+F)F+(3)1/2
(E-26F)S}+6S2]/6 -1
X(13349) [{(E+F)(2E-F)- (3)1/2
(E-5F)S}+2S2]/4 1/4
X(14538) [{(E-2F){2(E+F)+(3)1/2
S}-2S2]/2 1/2
X(16002) [{-(E+F)(E-2F)-12(3)1/2
FS}+6S2]/4 -3/4
Table 4 Euler coordinates (xe(n), ye(n)) of X(n) in N=$a$-class
X(n) xe(n)*$a$ ye(n)*$a$/{-$aSA(SB-SC)$}
X(1) $aSA$ 1
X(8) (E+F)$a$-2$aSA$ - 2
X(10) {(E+F)$a$-$aSA$}/2 - 1/2
X(40) (E-2F)$a$-$aSA$ - 1
X(145) -(E+F)$a$+4$aSA$ 4
X(165) {2(E-2F)$a$-$aSA$}/3 - 1/3
X(355) {(E+4F)$a$-2$aSA$}/2 - 1
X(551) {(E+F)$a$+3$aSA$}/6 1/2
X(944) -3F$a$+2$aSA$ 2
X(946) {3F$a$+$aSA$}/2 1/2
X(962) (-E+5F)$a$+2$aSA$ 2
X(1125) {(E+F)$a$+$aSA$}/4 1/4
X(1385) {(E-2F)$a$+2$aSA$}/4 1/2
X(1482) {(-E+2F)$a$+4$aSA$}/2 2
X(1483) {-(E+4F)$a$+8$aSA$}/4 2
X(1698) {2(E+F)$a$-$aSA$}/5 - 1/5
X(1699) {6F$a$+$aSA$}/3 1/3
X(3241) {-(E+F)$a$+6$aSA$}/3 2
X(3244) {-(E+F)$a$+5$aSA$}/2 5/2
X(3576) {(E-2F)$a$+$aSA$}/3 1/3
X(3579) {3(E-2F)$a$-2$aSA$}/4 - 1/2
X(3616) {(E+F)$a$+2$aSA$}/5 2/5
X(3617) {3(E+F)$a$-4$aSA$}/5 - 4/5
X(3621) 3(E+F)$a$-8$aSA$ - 8
X(3622) {(E+F)$a$+4$aSA$}/7 4/7
X(3623) {-(E+F)$a$+8$aSA$}/5 8/5
X(3624) {2(E+F)$a$+$aSA$}/7 1/7
X(3625) {3(E+F)$a$-7$aSA$}/2 - 7/2
X(3626) {3(E+F)$a$-5$aSA$}/4 - 5/4
X(3632) 2(E+F)$a$-5$aSA$ - 5
X(3633) -2(E+F)$a$+7$aSA$ 7
X(3634) {3(E+F)$a$-$aSA$}/8 - 1/8
X(3635) {-(E+F)$a$+7$aSA$}/4 7/4
X(3636) {(E+F)$a$+5$aSA$}/8 5/8
X(3653) {(5E-4F)$a$+6$aSA$}/18 1/3
X(3654) {(5E-4F)$a$-6$aSA$}/6 - 1
X(3655) {(E-8F)$a$+6$aSA$}/6 1
X(3656) {-(E-8F)$a$+6$aSA$}/6 1
X(3679) {2(E+F)$a$-3$aSA$}/3 - 1
X(3817) {(E+7F)$a$+$aSA$}/6 1/6
X(3828) {5(E+F)$a$-3$aSA$}/12 - 1/4
X(4297) {(E-5F)$a$+$aSA$}/2 1/2
X(4301) {-(E-5F)$a$+3$aSA$}/2 3/2
X(4668) {4(E+F)$a$-7$aSA$}/5 - 7/5
X(4669) {5(E+F)$a$-9$aSA$}/6 - 3/2
X(4677) {4(E+F)$a$-9$aSA$}/3 - 3
X(4678) {5(E+F)$a$-8$aSA$}/7 - 8/7
X(4691) {5(E+F)$a$-7$aSA$}/8 - 7/8
X(4701) {5(E+F)$a$-11$aSA$}/4 - 11/4
X(4745) {7(E+F)$a$-9$aSA$}/12 - 3/4
X(4746) {7(E+F)$a$-13$aSA$}/8 - 13/8
X(4816) {6(E+F)$a$-13$aSA$}/5 - 13/5
X(5493) {3(E-3F)$a$-3$aSA$}/2 - 3/2
X(5550) {3(E+F)$a$+2$aSA$}/11 2/11
X(5587) {(E+4F)$a$-$aSA$}/3 - 1/3
X(5603) {3F$a$+2$aSA$}/3 2/3
X(5657) {(2E-F)$a$-2$aSA$}/3 - 2/3
X(5690) {3E$a$-4$aSA$}/4 - 1
X(5691) 6F$a$-$aSA$ - 1
X(5731) {(E-5F)$a$+2$aSA$}/3 2/3
X(5734) {-(E-5F)$a$+6$aSA$}/5 6/5
X(5790) {3(E+2F)$a$-4$aSA$}/6 - 2/3
X(5818) {(2E+5F)$a$-2$aSA$}/5 - 2/5
X(5881) (E+4F)$a$-3$aSA$ - 3
X(5882) {-3F$a$+3$aSA$}/2 3/2
X(5886) {(E+4F)$a$+2$aSA$}/6 1/3
X(5901) {(E+4F)$a$+4$aSA$}/8 1/2
X(6361) (2E-7F)$a$-2$aSA$ - 2
X(6684) {(2E-F)$a$+-$aSA$}/4 - 1/4
X(7967) {-3F$a$+4$aSA$}/3 4/3
X(7982) -(E-2F)$a$+3$aSA$ 3
X(7987) {2(E-2F)$a$+$aSA$}/5 1/5
X(7988) {2(E+4F)$a$+$aSA$}/9 1/9
X(7989) {2(E+4F)$a$-$aSA$}/7 - 1/7
X(7991) 2(E-2F)$a$-3$aSA$ - 3
X(8148) {-3(E-2F)$a$+8$aSA$}/2 4
X(8227) {(E+4F)$a$+$aSA$}/5 1/5
X(9588) {2(2E-F)$a$-3$aSA$}/7 - 3/7
X(9589) -2(E-5F)$a$+3$aSA$ 3
X(9624) {(E+4F)$a$+3$aSA$}/7 3/7
X(9778) {3(E-3F)$a$-2$aSA$}/3 - 2/3
X(9780) {3(E+F)$a$-2$aSA$}/7 - 2/7
X(9812) {-(E-11F)$a$+2$aSA$}/3 2/3
X(9955) {(E+10F)$a$+2$aSA$}/8 1/4
X(9956) {3(E+2F)$a$-2$aSA$}/8 - 1/4
X(10164) {3(E-F)$a$-$aSA$}/6 - 1/6
X(10165) {(2E-F)$a$+$aSA$}/6 1/6
X(10171) {3(E+3F)$a$+$aSA$}/12 1/12
X(10172) {(4E+7F)$a$-$aSA$}/12 - 1/12
X(10175) {(2E+5F)$a$-$aSA$}/6 - 1/6
X(10222) {(-E+2F)$a$+6$aSA$}/4 3/2
X(10246) {(E-2F)$a$+4$aSA$}/6 2/3
X(10247) {-(E-2F)$a$+8$aSA$}/6 4/3
X(10248) {-E+23F)$a$+2$aSA$}/7 2/7
X(10283) {(E+4F)$a$+8$aSA$}/12 2/3
X(10595) {3F$a$+4$aSA$}/5 4/5
X(11224) {-2(E-2F)$a$+7$aSA$}/3 7/3
X(11230) {3(E+2F)$a$+2$aSA$}/12 1/6
X(11231) {(5E+2F)$a$-2$aSA$}/12 - 1/6
X(11278) {-3(E-2F)$a$+10$aSA$}/4 5/2
X(11362) {(2E-F)$a$-3$aSA$}/2 - 3/2
X(11522) {6F$a$+3$aSA$}/5 3/5
X(11531) -2(E-2F)$a$+5$aSA$ 5
X(12245) (2E-F)$a$-4$aSA$ - 4
X(12512) {3(E-3F)$a$-$aSA$}/4 - 1/4
X(12571) {(E+13F)$a$+$aSA$}/8 1/8
X(12645) {3(E+2F)$a$-8$aSA$}/2 - 4
X(12699) {-(E-8F)$a$+2$aSA$}/2 1
X(12702) {3(E-2F)$a$-4$aSA$}/2 - 2
X(13607) {-3F$a$+5$aSA$}/4 5/4
X(13464) {3F$a$+3$aSA$}/4 3/4
X(13624) {3(E-2F)$a$+2$aSA$}/8 1/4
X(15178) {(E-2F)$a$+6$aSA$}/8 3/4
X(15808) {3(E+F)$a$+5$aSA$}/14 5/14
X(16189) {-2(E-2F)$a$+9$aSA$}/5 9/5
X(16191) {-4(E-2F)$a$+17$aSA$}/9 17/9
X(16192) {4(E-2F)$a$-$aSA$}/7 - 1/7
X(16200) {-(E-2F)$a$+5$aSA$}/3 5/3
Table 5 Euler coordinates (xe(n), ye(n)) of X(n) in N=E-class
X(n) xe(n)*E ye(n)*E/(SB-SC)(SC-SA)(SA-SB)
X(51) {2(E+F)F+2S2}/3 - 1/3
X(52) {-(E2-4F
2)+4S
2}/2 - 1
X(143) {-(E2-4EF-8F
2)+8S
2}/8 - 1/2
X(185) -2(2E-F)F+2S2 - 1
X(373) {2(E+F)2+2S
2}/9 - 1/9
X(389) {-(E-2F)F+2S2}/2 - 1/2
X(568) {-(E2+2EF-8F
2)+8S
2}/6 - 2/3
X(1216) {(3E2+2EF-4F
2)-4S
2}/4 1/2
X(2979) {(3E2-EF-4F
2)-4S
2}/3 2/3
X(3060) {-(E2-3EF-4F
2)+4S
2}/3 - 2/3
X(3567) {(E+4F)F+4S2}/5 - 2/5
X(3819) {(3E2+EF-2F
2)-2S
2}/6 1/6
X(3917) {2(E2-F
2)-3S
2}/3 1/3
X(5446) {-(E2-6EF-4F
2)+4S
2}/4 - 1/2
X(5447) {(5E2-2EF-4F
2)-4S
2}/8 1/4
X(5462) {(E2+2EF+5F
2)+4S
2}/8 - 1/4
X(5562) (E2+2EF-2F
2)-2S
2 1
X(5640) {(E2+5EF+4F
2)+4S
2}/9 - 2/9
X(5650) {(4E2+2EF-2F
2)-2S
2}/3 1/3
X(5876) {(3E2+15EF-8F
2)-8S
2}/4 1
X(5889) -(E2+3EF-4F
2)+4S
2 - 2
X(5890) {-(5E-4F)F+4S2}/3 - 2/3
X(5891) {(3E2+8EF-4F
2)-4S
2}/6 1/3
X(5892) {(3E2-2EF+4F
2)+4S
2}/12 - 1/6
X(5907) {(E2+5EF-2F
2)-2S
2}/2 1/2
X(5943) {(E2+3EF+2F
2)+2S
2}/6 - 1/6
X(5946) {(E2+8F
2)+8S
2}/12 - 1/3
X(6101) {(5E2-8F
2)-8S
2}/4 1
X(6102) {-(E2+8EF-8F
2)+8S
2}/4 - 1
X(6146) -3(E+2F)F+2S2 - 1
X(6241) -(11E-4F)F+4S2 - 2
X(6243) {-(3E2-2EF-8F
2)+8S
2}/2 - 2
X(6688) {(3E2+5EF+2F
2)+2S
2}/12 - 1/12
X(7998) {(5E2+EF-4F
2)-4S
2}/9 2/9
X(7999) {(4E2+3EF-4F
2)-4S
2}/7 2/7
X(9729) {(E2-3EF+2F
2)+2S
2}/4 - 1/4
X(9730) {(E2-4EF+4F
2)+4S
2}/6 - 1/3
X(9781) {(7E+4F)F+4S2}/7 - 2/7
X(10095) {(E2+12EF+8F
2)+8S
2}/16 - 1/4
X(10110) {(5E+2F)F+2S2}/4 - 1/4
X(10112) {-(E2+5EF+10F
2)+6S
2}/2 - 3/2
X(10116) {-(E2+18EF+20F
2)+12S
2}/4 - 3/2
X(10170) {(5E2+10EF-4F
2)-4S
2}/12 1/6
X(10219) {(7E2+9EF+2F
2)+2S
2}/24 - 1/24
X(10263) {-(3E2-8EF-8F
2)+8S
2}/4 - 1
X(10574) {(E2-7EF+4F
2)+4S
2}/5 - 2/5
X(10575) {(E2-16EF+4F
2)+4S
2}/2 - 1
X(10625) {(3E2-4EF-4F
2)-4S
2}/2 1
X(10627) {(7E2-4EF-8F
2)-8S
2}/8 1/2
X(11002) {-(E2-7EF-8F
2)+8S
2}/9 - 4/9
X(11017) {(9E2+52EF-8F
2)-8S
2}/32 1/8
X(11225) {-(E2+7EF+6F
2)+10S
2}/5 - 1
X(11245) {-4(E+F)F+4S2}/3 - 2/3
X(11264) {-(3E2+28EF+40F
2)+24S
2}/8 - 3/2
X(11381) 2(5E-F)F-2S2 1
X(11412) (2E2+EF-4F
2)-4S
2 2
X(11439) {(E2+21EF-4F
2)-4S
2}/5 2/5
X(11444) {(3E2+5EF-4F
2)-4S
2}/5 2/5
X(11451) {(3E2+7EF+4F
2)+4S
2}/15 - 2/15
X(11455) {(23E-4F)F-4S2}/3 2/3
X(11459) {(2E2+7EF-4F
2)-4S
2}/3 2/3
X(11465) {(4E2+5EF+4F
2)+4S
2}/17 - 2/17
X(11565) {(3E2-44EF-104F
2)+24S
2}/16 - 3/4
X(11591) {(5E2+12EF-8F
2)-8S
2}/8 1/2
X(11592) {(17E2-12EF-8F
2)-8S
2}/32 1/8
X(11695) {(2E2+EF+2F
2)+2S
2}/8 - 1/8
X(11754) {(4E+9F)F+2S2}/4 - 1/4
X(11750) {(E2-12EF-28F
2)+4S
2}/2 - 1
X(11793) {(2E2+3EF-2F
2)-2S
2}/4 1/4
X(12002) {-(3E2-30EF-12F
2)+12S
2}/16 - 3/8
X(12006) {(3E2-4EF+8F
2)+8S
2}/16 - 1/4
X(12022) {-3(E+4F)F+4S2}/3 - 2/3
X(12024) {-3(2E+5F)F+5S2}/3 - 5/6
X(12045) {(11E2+13EF+2F
2)+2S
2}/36 - 1/36
X(12046) {(15E2+44EF+8F
2)+8S
2}/64 - 1/16
X(12111) (E2+9EF-4F
2)-4S
2 2
X(12134) {(E2+10EF+12F
2)-4S
2}/2 1
X(12162) {(E2+12EF-4F
2)-4S
2}/2 1
X(12241) -3F2+S
2 - 1/2
X(12278) (E2+5EF+28F
2)-4S
2 2
X(12279) (E2-19EF+4F
2)+4S
2 - 2
X(12289) -(7E-32F)F+4S2 - 2
X(12290) (17E-4F)F-4S2 2
X(12370) {-(E2+4EF+24F
2)+8S
2}/4 - 1
X(12897) {-(E2-10EF+28F
2)+4S
2}/4 - 1/2
X(13142) {-(E2-EF+4F
2)+4S
2}/2 - 1
X(13292) {-(E2+6EF-4F
2)+8S
2}/4 - 1
X(13340) {(7E2-10EF-8F
2)-8S
2}/6 2/3
X(13348) {(3E2-5EF-2F
2)-2S
2}/4 1/4
X(13363) {(5E2+4EF+8F
2)+8S
2}/24 - 1/6
X(13364) {(3E2+20EF+8F
2)+8S
2}/24 - 1/6
X(13382) {-3(3E-2F)F+6S2}/4 - 3/4
X(13403) {(E-14F)F+2S2}/2 - 1/2
X(13419) {(11E+14F)F-2S2}/2 1/2
X(13421) {-(9E2-12EF-24F
2)+24S
2}/8 - 3/2
X(13451) {-(E2-24EF-16F
2)+16S
2}/24 - 1/3
X(13470) {(3E2-20EF-56F
2)+8S
2}/8 - 1/2
X(13474) {(13E-2F)F-2S2}/2 1/2
X(13491) {(E2-24EF+8F
2)+8S
2}/4 - 1
X(13568) -(E-5F)F+S2 - 1/2
X(13570) {(E2+21EF+2F
2)+2S
2}/12 - 1/12
X(13598) {-(E2-7EF-2F
2)+2S
2}/2 - 1/2
X(13630) {(E2-12EF+8F
2)+8S
2}/8 - 1/2
X(14128) {(7E2+20EF-8F
2)-8S
2}/16 1/4
X(14449) {-(5E2-8EF-16F
2)+16S
2}/12 - 1
X(14516) (E2+7EF+12F
2)-4S
2 2
X(14531) -2(E2+EF-3F
2)+6S
2 - 3
X(14641) {(3E2-26EF+4F
2)+4S
2}/4 - 1/2
X(14831) {-(E2+4EF-6F
2)+6S
2}/3 - 1
X(14845) {(3E2+16EF+4F
2)+4S
2}/18 - 1/9
X(14855) {(3E2-20EF+4F
2)+4S
2}/6 - 1/3
X(15012) {(E2-5EF+6F
2)+6S
2}/8 - 3/8
X(15028) {(3E2+EF+4F
2)+4S
2}/13 - 2/13
X(15030) {(E2+8EF-2F
2)-2S
2}/3 1/3
X(15043) (E2-EF+4F
2)+4S
2 - 2/7
X(15060) {(5E2+24EF-8F
2)-8S
2}/12 1/3
X(15067) {(7E2+8EF-8F
2)-8S
2}/12 - 1/3
X(15072) {(E2-13EF+4F
2)+4S
2}/3 - 2/3
X(15305) {(E2+15EF-4F
2)-4S
2}/3 2/3
X(15606) {(4E2+EF-6F
2)-6S
2}/4 3/4
X(15644) {(2E2-3EF-2F
2)-2S
2}/2 1/2
X(15807) {(E2+20EF-56F
2)+8S
2}/16 - 1/4
X(16194) {(E2+24EF-4F
2)-4S
2}/6 1/3
X(16226) {(E2-2EF+6F
2)+6S
2}/9 - 1/3
Table 6 Euler coordinates (xe(n), ye(n)) of X(n) in N =F -class
X(n) xe(n)*F ye(n)*F/{(SB-SC)(SC-SA)(SA-SB)}
X(64) 2(2E-F)F-S2 1/2
X(68) {(E+2F)(E+4F)-4S2}/4 1/2
X(154) {-2(E+F)F+S2}/3 - 1/6
X(156) {-(E2+12EF+8F
2)+8S
2}/16 - 1/4
X(1147) {-(E2+2EF+4F
2)+4S
2}/8 - 1/4
X(1498) -3EF+S2 - 1/2
X(1853) {4(E+F)F-S2}/3 1/6
X(2883) {-3(E-F)F+S2}/2 - 1/4
X(3167) {-(E+F)(E+2F)+4S2}/6 - 1/3
X(3357) {3(3E-2F)F-2S2}/4 1/4
X(5448) {-(E2+2EF-20F
2)+4S
2}/16 - 1/8
X(5449) {(E2+10EF+12F
2)-4S
2}/16 1/8
X(5654) {-(E+4F)(E-2F)+4S2}/12 - 1/6
X(5656) {-3(2E-F)F+2S2}/3 - 1/3
X(5878) {-(7E-8F)F+2S2}/2 - 1/2
X(5893) {-6(E-3F)F+2S2}/8 - 1/8
X(5894) {(5E-7F)F-S2}/2 1/4
X(5895) -4(E-2F)F+S2 - 1/2
X(5925) 5(E-2F)F-S2 1/2
X(5944) {-(3E2+16EF+28F
2)+16S
2}/28 - 2/7
X(6193) {-(E2+4EF+6F
2)+4S
2}/2 - 1
X(6225) -(7E-5F)F+2S2 - 1
X(6247) {(4E+F)F-S2}/2 1/4
X(6696) {(5E-F)F-S2}/4 1/8
X(6759) {-(5E+2F)F+2S2}/4 - 1/4
X(7689) {(E2+10EF-12F
2)-4S
2}/8 1/4
X(8567) {6(E-F)F-S2}/5 1/10
X(9820) {-(E2-4F
2)+4S
2}/16 1/8
X(9833) {-(5E+8F)F+2S2}/2 - 1/2
X(9927) {(E2+6EF+20F
2)-4S
2}/8 1/4
X(9936) {-(3E2+14EF+8F
2)+12S
2}/4 - 3/2
X(10182) {3(E-2F)F+2S2}/24 - 1/24
X(10192) {-(E+F)F+S2}/6 - 1/12
X(10193) {(17E-10F)F-2S2}/24 1/24
X(10282) {-3(E+2F)F+2S2}/8 - 1/8
X(10606) {(5E-4F)F-S2}/3 1/6
X(11202) {-(E+10F)F+2S2}/12 - 1/12
X(11204) {(13E-14F)F-2S2}/12 1/12
X(11206) {-5(E+F)F+2S2}/3 - 1/3
X(11411) {(E2+6EF+2F
2)-4S
2}/2 1
X(12038) {-(E2-2EF+12F
2)+4S
2}/16 - 1/8
X(12118) {-(E2+2EF+16F
2)+4S
2}/4 - 1/2
X(12163) {(E2+8EF-8F
2)-4S
2}/4 1/2
X(12164) {-(E2+5EF-2F
2)+4S
2}/2 - 1
X(12250) (8E-7F)F-2S2 1
X(12293) {(E2+4EF+24F
2)-4S
2}/4 1/2
X(12315) {-(13E-2F)F+4S2}/2 - 1
X(12324) (7E+F)F-2S2 1
X(12359) {(7E+4F)F-2S2}/8 1/4
X(12429) {(E2+5EF+10F
2)-4S
2}/2 1
X(13093) {3(5E-2F)F-4S2}/2 1
X(13346) {-(E2-3EF+2F
2)+2S
2}/4 - 1/4
X(13561) {(E+28F)E-8S2}/32 1/8
X(14216) {(7E+4F)F-2S2}/2 1/2
X(14530) {3(3E+2F)F+4S2}/10 - 1/5
X(14852) {(E+4F)2-4S
2}/12 1/6
X(14862) {-3(5E-2F)F+6S2}/16 - 3/16
X(14864) {3(7E+6F)F-6S2}/8 3/8
X(16252) {-(2E-F)F+S2}/4 - 1/8
Table 7 Euler coordinates (xe(n), ye(n)) of X(n) in N = (E-8F)-class