New Simpler Equations for Properties of
Hypoellipse ,Ellipse and Superellipse Curves
Maher Izzedin Aldaher; B sc. C. E. ; Mosul Univ. 1987, C. E. Municipality of Irbid West
email : maher_daher2000@yahoo.com, Mobile no.: +962(0)795133967,+962(0)776733564
:‫الملخص‬
‫المساحة‬،‫الزاويا‬،‫هذه الدراسة تعطي حل مبتكر تقريبي بشكل مبسط وعملي اليجاد الجذور التريعية‬
‫فوق‬،‫القطع الناقص‬،‫المحيط وطول القوس لمنحنى تحت القطع الناقص‬،‫الكاملة والجزئية‬
‫ فوق القطع‬،‫منحنى المي‬،‫ القطع الناقص المستطيل‬،‫(فوق القطع الناقص‬،‫الدائرة المربعة‬،‫الدائري‬
‫ الصفات األخرى ذات العالقة مثل نصف القطر‬، ‫الناقص لبيي هين) في المستوى ذو االتجاهين‬
. ‫الهيدروليكي يمكن إيجادها بسهولة‬
Abstract:
This study gives a novel approximate solution in a simple practical
form to find square roots, angles, full and partial area , perimeter ,arc
length for the curve of Hypoellipse ,Ellipse , (Hypercircle, Supercircle),
Squircle ,(Hyperellipse, Superellipse, Rectellipse, Hyperoval
,Superoval ,Lame curve,Peit Hein Superellipse) in 2D plan, other
related properties such as Hydraulic radius found easily.
Symbols:
a : major radius on x axis
A : area in 1st quadrant
b : minor radius on y axis
Ax : partial area
n : power varies from 1 to ∞
Pw : wetted perimeter
P : perimeter in 4 quadrants
Rh : hydraulic radius
L : arc length in 1st quadrant
Lx : partial arc length
At : total area in 4 quadrants
Section 1: Introduction
This study started from replying a question "What's the
perimeter of Ellipse?",in1988,the answer I've found in a simple
form wrote in my non printed book "The Elliptical Shape" in
1992 ,which registered in the National Library ,Press
Department-Jordan under the serial no.446/8/1992,then
followed the research in 1996 to obtain the approximations
of the complete area and arc length, perimeter for the
𝒙𝒏
𝒚𝒏
general function 𝒂𝒏 + 𝒃𝒏 = 𝟏, then for partial area in2004.
Several helpful tools used in the research, such as
theoretical scientific base from what studied in mathematics in
general and especially in calculus, statics, engineering
mathematics, with other special subjects such as special
functions, drawing ,also the collected knowledge from reference
books, researches and web sites with the aid of programmable
calculator and computer programs.
It's an applicable research for different sectors such as
applied mathematics, statistics , physics, hydraulics, heat transfer,
aerodynamic , mechatronics, antennas, botany, live beings,
medical images, computer vision, machineries, technicians,
designer engineers, etc.. .Equations can be used to approximate
the complete and incomplete elliptic integrals of second kind,
gamma and beta functions also hypergeometric functions.
Superquadrics as 3D bodies and special cases of Superformula in
2D & 3D plans can be studied. Results of equations gives good
approximations as indicated in the tables. References listed at the
end which make an obvious image about the subject .
Section 2: Basic Concepts
𝑎 ≥ 𝑏 ≥ 0
∞≥n≥ 1
𝑥𝑛
𝑦𝑛
𝑎
𝑏𝑛
𝑛 +
𝑥
= 1 ; f(x) = y = b(1 − ( )𝑛 )1/𝑛
𝑎
n=1
( Rhombus, Diamond), Straight line in 1st quad.
2>n> 1
Hypoellipse
n=2
Ellipse
n=2 , a=b
Circle
n>2 , a=b
(Hypercircle, Supercircle)
n=4 , a=b
Squircle
n>2 , a>b
(Hyperellipse, Superellipse, Rectellipse,
Hyperoval, Superoval, Lame curve)
n=2.5 , a>b
Peit Hein Superellipse
n=∞ , a=b
Square
n=∞ , a>b
Rectangle
fig.(1-2)
Area under the curve :
𝑥
Ax = ∫0 𝑓(𝑥 )𝑑𝑥
=
𝑥
∫0 b(1
𝑥 𝑛 1/𝑛
−( ) )
𝑎
𝑑𝑥
Arc length of the curve :
𝑥
Lx = ∫0 √1 + 𝑓 ′ (𝑥)2
𝑓 ′ (𝑥) =
Lx =
−𝑏
𝑥
∫0 √1
𝑎
𝑥
𝑑𝑥
1
(𝑛)−1 𝑥 𝑛−1
( )
𝑎
(1 − ( )𝑛 )
𝑎
𝑏
+ (𝑎 )2 ((1 −
1
𝑥 𝑛 ( )−1 𝑥 𝑛−1
( ) ) 𝑛 ( ) )2
𝑎
𝑎
Hydraulic radius :
Rh =
𝐚𝐫𝐞𝐚
𝒘𝒆𝒕𝒕𝒆𝒅 𝒑𝒆𝒓𝒊𝒎𝒆𝒕𝒆𝒓
=
𝑨
𝑷
𝑑𝑥
Section 3: Square Roots
a≥b≥ 0
√𝒂𝟐
+
𝒃𝟐
𝟓𝒃𝟐
= a + 𝟗𝒂 +𝟑𝒃
…. eq.(1-3)
√𝒂𝟐 + 𝒃𝟐
b
fig.(1-3)
table (1-3)
a
1
1
1
1
1
1
1
1
1
1
1
b
1
.9
.8
.7
.6
.5
.4
.3
.2
.1
0
(a^2+b^2)^.5
1.4142136
1.3453624
1.2806248
1.2206556
1.1661904
1.118034
1.077033
1.0440307
1.0198039
1.0049876
1
eq.(1-3)
1.416667
1.346154
1.280702
1.220721
1.166667
1.119048
1.078431
1.045455
1.020833
1.005376
1
a
Section 4: Angles
𝜋
2
≥θ≥ 0
180 𝜋
θ =(
𝜋
2
) sin θ (1 − cos θ + 0.423cos2 θ − 0.15 cos 3 θ )
2
𝜋
………..eq.(1-4)
θ
=(
𝑛=4
180 𝜋
𝜋
) sin θ (1 − ∑𝑛=1(−1)𝑛 (𝑅𝑛)cos 𝑛 θ )
2
………..eq.(2-4)
𝑅1 =
θ
=(
2
; 𝑅2 = 0.47583; 𝑅3 = 0.29178 ; 𝑅4 = 0.09004
𝜋
4
1−cos θ+0.1416 sin θ
)(sin θ + (1 − cos θ ) ⁄3 (
))
180
𝜋
1−cos θ+sin θ
………..eq.(3-4)
table (1-4)
𝐬𝐢𝐧 𝛉
0.000
0.342021
0.50000
0.707108
0.866027
0.984808
1.000
𝐜𝐨𝐬 𝛉
1.000
0.939692
0.866025
0.707105
0.5 0000
0.173645
0.000
θ
0
20
30
45
60
80
90
eq.(1-4)
0.000
20.033697
30.082325
45.07667
59.913662
79.895652
90.00021
eq.(2-4)
0.000
20.00944
30.00048
45.00019
60.00037
79.9783
90.00021
eq.(3-4)
0.000
19.96253
29.91656
44.89428
59.98335
80.11747
90.00004
Section 5: Perimeter & Arc Length of Ellipse
a
fig.(1-5)
a≥b≥ 0
a≥b
Ellipse ; a = b Circle
𝑥2
𝑦2
𝑎
𝑏2
2 +
𝑥
= 1 ; f(x) = y = b(1 − ( )2 )1/2
P=4(a+b-
𝑎
0.8584𝑎𝑏
Lx=x+(b-y)(
𝑎+𝑏
)
………..eq.(1-5)
b−y+0.1416x
b−y+x
𝑏−𝑦 1/3
)
𝑏
)(
………..eq.(2-5)
table (1-5)
a
b
n
P
1
1
2
6.283
1
1
2
1
1
1
1
x
θ
y
Lx
L(a-x)
0.2 11.54
0.98
0.201212
1.37
6.283
0.5
0.866 0.522137
1.049
2
6.283
0.7 44.43 0.714
2
6.283
1
30
90
0
0.77352
0.797
1.57075
0
1 0.8 2 5.673778 0.2 11.54 0.784 0.200906
1.218
1 0.8 2 5.673778 0.5
0.693 0.516071
0.902
1 0.8 2 5.673778 0.7 44.43 0.571 0.753163
0.665
1 0.8 2 5.673778
0
1.418444
0
1 0.5 2 4.855333 0.2 11.54
0.49
0.200503
1.013
1 0.5 2 4.855333 0.5
0.433 0.508327
0.706
1 0.5 2 4.855333 0.7 44.43 0.357 0.727028
0.487
1 0.5 2 4.855333
1
30
1
90
30
90
0
1.213833
0
1 0.3 2 4.407538 0.2 11.54 0.294 0.200275
0.902
1 0.3 2 4.407538 0.5
0.504224
0.598
1 0.3 2 4.407538 0.7 44.43 0.214 0.713287
0.389
1 0.3 2 4.407538
1
30
90
0.26
0
1.101885
0
1 0.1 2 4.087818 0.2 11.54 0.098 0.200083
0.822
1 0.1 2 4.087818 0.5
0.087 0.501124
0.521
1 0.1 2 4.087818 0.7 44.43 0.071 0.703299
0.319
1 0.1 2 4.087818
1
30
90
0
1.021955
0
Section 6:Area&Perimeter for
(
𝒙𝒏
𝒂𝒏
+
𝒚𝒏
𝒃𝒏
= 𝟏)
L=a+b*(((2.5/(n+0.5))^(1/n))*b+a*(n1)*0.566/n^2)/(b+a*(4.5/(0.5+n^2)))
………..eq.(1-6)
Lx=x+(b-y)*((((2.5/(n+0.5))^(1/n))*(b-y)+0.566*x*(n-1)/(n^2))/(by+x*4.5/(n^2+0.5)))*((b-y)/b)^((n-1)/3)
………..eq.(2-6)
P=L*4
………..eq.(3-6)
L(a-x)=L-Lx
………..eq.(4-6)
A=a*b*((0.5)^((n^(-1.52))))
………..eq.(5-6)
At=A*4
………..eq.(6-6)
A(a-x) =a*y*((0.5^(n^(-1.52)))-(n^4/((n^4+1)))*(x/a)+(n1)*((1/(n+1))^(n-1))*(((x/a)^(2*n-1)))-(n-1)*((0.117)^(n1))*(((x/a)^(2*n-1))))
………..eq.(7-6)
Ax=A- A(a-x)
………..eq.(8-6)
table (1-6)
a b n L Pmaher x
θ
y Lxmaher LxExcel At A(a-x) A(x)maher A(x)mathm
1 1
1 1.417 5.666667 0.2 11.54 0.8
0.283333
0.2828427
0.5
0.32
0.18
0.18
1 1
1 1.417 5.666667 0.5 30
0.5
0.708333
0.7071068
0.5
0.125
0.375
0.375
1 1
1 1.417 5.666667 0.7 44.43 0.3
0.991667
0.9899495
0.5
0.045
0.455
0.455
1 1
1 1.417 5.666667
1.416667
1.4142136
0.5
0
0.5
0.5
1 0.5 1 1.119 4.47619 0.2 11.54 0.4
0.22381
0.2236068 0.25
0.16
0.09
0.09
1 0.5 1 1.119 4.47619 0.5 30
0.25
0.559524
0.559017
0.25
0.063
0.1875
0.1875
1 0.5 1 1.119 4.47619 0.7 44.43 0.15
0.783333
0.7826238 0.25
0.023
0.2275
0.2275
1 0.5 1 1.119 4.47619
1.119048
1.118034
0
0.25
0.25
1 0.1 1 1.005 4.021505 0.2 11.54 0.08
0.201075
0.2009975 0.05
0.032
0.018
0.018
1 0.1 1 1.005 4.021505 0.5 30
0.05
0.502688
0.5024938 0.05
0.013
0.0375
0.0375
1 0.1 1 1.005 4.021505 0.7 44.43 0.03
0.703763
0.7034913 0.05
0.005
0.0455
0.0455
1 0.1 1 1.005 4.021505
1.005376
1.0049876 0.05
0
0.05
0.05
1 1 1.3 1.454 5.817424 0.2 11.54 0.904
0.221667
0.2230381 0.628 0.439
0.18913855
0.191667
1 1 1.3 1.454 5.817424 0.5 30
0.67
0.603516
0.6039063 0.628 0.189
0.4386266
0.429746
1 1 1.3 1.454 5.817424 0.7 44.43 0.466
0.892441
0.889248 0.628 0.071
0.55687084
0.54407
1 1 1.3 1.454 5.817424
1.454356
1.4478809 0.628
0.62801497
0.624321
1 0.5 1.3 1.146 4.582606 0.2 11.54 0.452
0.206819
0.2060385 0.314 0.219
0.09456928
0.0958337
1 0.5 1.3 1.146 4.582606 0.5 30 0.335
0.532575
0.5282185 0.314 0.095
0.2193133
0.214873
1 0.5 1.3 1.146 4.582606 0.7 44.43 0.233
0.760805
0.7526428 0.314 0.036
0.27843542
0.272035
1 0.5 1.3 1.146 4.582606
1.145651
1.1361172 0.314
0.31400748
0.31216
1 0.1 1.3 1.011 4.042551 0.2 11.54 0.09
0.200589
0.2002456 0.063 0.044
0.01891386
0.0191667
1 0.1 1.3 1.011 4.042551 0.5 30 0.067
0.502586
0.5011654 0.063 0.019
0.04386266
0.0429746
1 0.1 1.3 1.011 4.042551 0.7 44.43 0.047
0.70467
0.7022001 0.063 0.007
0.05568708
0.054407
1 0.1 1.3 1.011 4.042551
1.010638
1.0061342 0.063
0.0628015
0.0624321
1 1 1.5 1.488 5.951438 0.2 11.54 0.939
0.209337
0.2100188 0.688 0.495
0.19309032
0.195184
1 1 1.5 1.488 5.951438 0.5 30 0.748
0.566664
0.5670252 0.688 0.229
0.45859524
0.450885
1
1
1
1
1
1
90
90
90
90
90
90
0
0
0
0
0
0
0.25
0
0
0
1 1 1.5 1.488 5.951438 0.7 44.43 0.556
0.847314
0.8444525 0.688 0.097
0.59086579
0.582231
1 1 1.5 1.488 5.951438
1.487859
1.4847092 0.688
0.68780201
0.684463
1 0.5 1.5 1.165 4.660914 0.2 11.54 0.47
0.203199
0.2025665 0.344 0.247
0.09654516
0.097592
1 0.5 1.5 1.165 4.660914 0.5 30 0.374
0.522233
0.517858 0.344 0.115
0.22929762
0.225442
1 0.5 1.5 1.165 4.660914 0.7 44.43 0.278
0.749055
0.7398121 0.344 0.048
0.29543289
0.291116
1 0.5 1.5 1.165 4.660914
1.165229
1.1575142 0.344
0.34390101
0.342232
1 0.1 1.5 1.014 4.055707 0.2 11.54 0.094
0.200366
0.2001035 0.069 0.049
0.01930903
0.0195184
1 0.1 1.5 1.014 4.055707 0.5 30 0.075
0.502192
0.5007315 0.069 0.023
0.04585952
0.0450885
1 0.1 1.5 1.014 4.055707 0.7 44.43 0.056
0.704552
0.7016571 0.069
0.01
0.05908658
0.0582231
1 0.1 1.5 1.014 4.055707
1.013927
1.0079951 0.069
0
0.0687802
0.0684463
1 1 1.7 1.522 6.088462 0.2 11.54 0.961
0.204111
0.2044431 0.734 0.538
0.19617391
0.197152
1 1 1.7 1.522 6.088462 0.5 30 0.805
0.543065
0.5437939 0.734 0.261
0.47293897
0.465008
1 1 1.7 1.522 6.088462 0.7 44.43 0.629
0.812291
0.8109467 0.734
0.12
0.61381816
0.609641
1 1 1.7 1.522 6.088462
1.522115
1.5212941 0.734
0
0.73387524
0.73174
1 0.5 1.7 1.185 4.73999 0.2 11.54 0.481
0.201533
0.2011248 0.367 0.269
0.09808696
0.0985758
1 0.5 1.7 1.185 4.73999 0.5 30 0.403
0.515133
0.5114951 0.367
0.13
0.23646949
0.232504
1 0.5 1.7 1.185 4.73999 0.7 44.43 0.314
0.73908
0.7302918 0.367
0.06
0.30690908
0.30482
1 0.5 1.7 1.185 4.73999
1.184997
1.1797608 0.367
0
0.36693762
0.36587
1 0.1 1.7 1.017 4.068627 0.2 11.54 0.096
0.200212
0.2000452 0.073 0.054
0.01961739
0.0197152
1 0.1 1.7 1.017 4.068627 0.5 30 0.081
0.50174
0.500468 0.073 0.026
0.0472939
0.0465008
1 0.1 1.7 1.017 4.068627 0.7 44.43 0.063
0.704142
0.7012553 0.073 0.012
0.06138182
0.0609641
1 0.1 1.7 1.017 4.068627
1.017157
1.0105798 0.073
0.07338752
0.073174
1
1
1
1
1
1
90
90
90
90
90
90
0
0
0
0
0
0
0
0
0
1 1
2 1.571
6.283
0.2 11.54 0.98
0.201212
0.2013573 0.785 0.587
0.19860273
0.198659
1 1
2 1.571
6.283
0.5 30 0.866
0.522137
0.5235979 0.785 0.296
0.48933272
0.478306
1 1
2 1.571
6.283
0.7 44.43 0.714
0.77352
0.7753974 0.785 0.143
0.6419864
0.637649
1 1
2 1.571
6.283
1.57075
1.5706394 0.785
0.78529906
0.785398
1 0.5 2 1.214 4.855333 0.2 11.54 0.49
0.200503
0.2003409 0.393 0.293
0.09930137
0.0993293
1 0.5 2 1.214 4.855333 0.5 30 0.433
0.508327
0.5060918 0.393 0.148
0.24466636
0.239153
1 0.5 2 1.214 4.855333 0.7 44.43 0.357
0.727028
0.720294 0.393 0.072
0.3209932
0.318824
1
90
0
0
1 0.5 2 1.214 4.855333
1
1.213833
1.2107791 0.393
0.39264953
0.392699
1 0.1 2 1.022 4.087818 0.2 11.54 0.098
0.200083
0.2000137 0.079 0.059
0.01986027
0.0198659
1 0.1 2 1.022 4.087818 0.5 30 0.087
0.501124
0.5002464 0.079
0.03
0.04893327
0.0478306
1 0.1 2 1.022 4.087818 0.7 44.43 0.071
0.703299
0.7008354 0.079 0.014
0.06419864
0.0637649
1 0.1 2 1.022 4.087818
1.021955
1.015305 0.079
0.07852991
0.0785398
1
90
90
0
0
0
0
1 1
3 1.692 6.767706 0.2 11.54 0.997
0.200015
0.2000322 0.878 0.678
0.19934827
0.199866
1 1
3 1.692 6.767706 0.5 30 0.956
0.501956
0.5034543 0.878
0.37
0.50769117
0.494661
1 1
3 1.692 6.767706 0.7 44.43 0.869
0.71491
0.722481 0.878 0.176
0.70152867
0.678471
1 1
3 1.692 6.767706
1.691927
1.6860646 0.878
0.87765896
0.883319
1 0.5 3 1.294 5.176417 0.2 11.54 0.499
0.200007
0.200008 0.439 0.339
0.09967414
0.0999332
1 0.5 3 1.294 5.176417 0.5 30 0.478
0.500858
0.50087
0.439 0.185
0.25384559
0.247331
1 0.5 3 1.294 5.176417 0.7 44.43 0.435
0.706207
0.7058256 0.439 0.088
0.35076433
0.339236
1 0.5 3 1.294 5.176417
1.294104
1.2860697 0.439
0.43882948
0.44166
1 0.1 3 1.038 4.150025 0.2 11.54 0.1
0.200001
0.2000003 0.088 0.068
0.01993483
0.0199866
1 0.1 3 1.038 4.150025 0.5 30 0.096
0.500149
0.5000349 0.088 0.037
0.05076912
0.0494661
1 0.1 3 1.038 4.150025 0.7 44.43 0.087
0.700974
0.700236 0.088 0.018
0.07015287
0.0678471
1 0.1 3 1.038 4.150025
0
1.037506
1.0313494 0.088
0
0.0877659
0.0883319
1
0.2
0.2000009 0.919
0.72
0.1995097
0.199984
1
1
1
90
90
90
0
0
0
0
1 1
4 1.762 7.04689 0.2 11.54
1 1
4 1.762 7.04689 0.5 30 0.984
0.500112
0.5005918 0.919 0.415
0.50464708
0.498417
1 1
4 1.762 7.04689 0.7 44.43 0.934
0.702251
0.7074463 0.919 0.209
0.7105212
0.691129
1 1
4 1.762 7.04689
1.761723
1.7542128 0.919
0
0.91917925
0.927037
1 0.5 4 1.348 5.39194 0.2 11.54 0.5
0.2
0.2000002 0.46
0.36
0.09975485
0.099992
1 0.5 4 1.348 5.39194 0.5 30 0.492
0.500053
0.5001482 0.46
0.207
0.25232354
0.249208
1 0.5 4 1.348 5.39194 0.7 44.43 0.467
0.701011
0.7018919 0.46
0.104
0.3552606
0.345565
1 0.5 4 1.348 5.39194
1.347985
1.3316356 0.46
0
0.45958962
0.463519
0.092 0.072
0.01995097
0.0199984
1
1
90
90
0
0
1 0.1 4 1.052 4.206541 0.2 11.54 0.1
0.2
1 0.1 4 1.052 4.206541 0.5 30 0.098
0.50001
0.5000059 0.092 0.041
0.05046471
0.0498417
1 0.1 4 1.052 4.206541 0.7 44.43 0.093
0.700178
0.7000761 0.092 0.021
0.07105212
0.0691129
1 0.1 4 1.052 4.206541
1.051635
1.0434287 0.092
0.09191792
0.0927037
1
90
0
0.2
0
1 1
8 1.86 7.440225 0.2 11.54
1
0.2
1 1
8 1.86 7.440225 0.5 30
1
1 1
8 1.86 7.440225 0.7 44.43 0.993
1 1
8 1.86 7.440225
1
90
0
0.971 0.771
0.19995143
0.2
0.5
0.500001 0.971 0.471
0.50010841
0.499973
0.7
0.700169 0.971 0.269
0.70183465
0.699432
1.8691588 0.971
0.97104234
0.978461
0.486 0.386
0.09997572
0.1
1.860056
0.2
1 0.5 8 1.431 5.723451 0.2 11.54 0.5
0.2
1 0.5 8 1.431 5.723451 0.5 30
0.5
0.5000003 0.486 0.235
0.25005421
0.249986
0.7
0.7000423 0.486 0.135
0.35091733
0.349716
1.4095861 0.486
0.48552117
0.48923
0.5
1 0.5 8 1.431 5.723451 0.7 44.43 0.496
1 0.5 8 1.431 5.723451
1
90
0
1.430863
0.2
0
0
1 0.1 8 1.087 4.348056 0.2 11.54 0.1
0.2
0.2
0.097 0.077
0.01999514
0.02
1 0.1 8 1.087 4.348056 0.5 30
0.5
0.5
0.097 0.047
0.05001084
0.0499973
0.7000017 0.097 0.027
0.07018347
0.0699432
1.0667961 0.097
0.09710423
0.0978461
0.1
1 0.1 8 1.087 4.348056 0.7 44.43 0.099
1 0.1 8 1.087 4.348056
1
0
1.087014
1 1 20 1.917 7.666893 0.2 11.54
1
0.2
0.2
0.993 0.793
0.19999875
0.2
1 1 20 1.917 7.666893 0.5 30
1
0.5
0.5
0.993 0.493
0.4999969
0.5
1 1 20 1.917 7.666893 0.7 44.43
1
0.7
0.7
0.993 0.293
0.70000731
0.699999
1 1 20 1.917 7.666893
0
1.916723
0.99272763
0.996174
1
90
0.7
90
1.945223 0.993
0
0
1 0.5 20 1.466 5.865887 0.2 11.54 0.5
0.2
0.2
0.496 0.396
0.09999938
0.1
1 0.5 20 1.466 5.865887 0.5 30
0.5
0.5
0.5
0.496 0.246
0.24999845
0.25
1 0.5 20 1.466 5.865887 0.7 44.43 0.5
0.7
0.7
0.496 0.146
0.35000365
0.349999
0.49636382
0.498087
1 0.5 20 1.466 5.865887
1
90
0
1.466472
1.4614597 0.496
0
1 0.1 20 1.105 4.420364 0.2 11.54 0.1
0.2
0.2
0.099 0.079
0.01999988
0.02
1 0.1 20 1.105 4.420364 0.5 30
0.1
0.5
0.5
0.099 0.049
0.04999969
0.05
1 0.1 20 1.105 4.420364 0.7 44.43 0.1
0.7
0.7
0.099 0.029
0.07000073
0.0699999
0.09927276
0.0996174
1 0.1 20 1.105 4.420364
1
90
0
1.105091
1.0829743 0.099
0
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