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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:
هذه الدراسة تعطي حل مبتكر تقريبي بشكل مبسط وعملي لايجاد الجذور التريعية،الزاويا،المساحة الكاملة والجزئية،المحيط وطول القوس لمنحنى تحت القطع الناقص،القطع الناقص،فوق الدائري،الدائرة المربعة،(فوق القطع الناقص، القطع الناقص المستطيل،منحنى لامي، فوق القطع الناقص لبيي هين) في المستوى ذو الاتجاهين ، الصفات الأخرى ذات العلاقة مثل نصف القطر 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 general function x^n/a^n +y^n/b^n =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
x^n/a^n +y^n/b^n =1 ; f(x) = y = b〖(1-〖(x^ /a^ )〗^n)〗^(1/n)
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 :
Ax = ∫_0^x▒f(x)dx
= ∫_0^x▒〖b〖(1-〖(x^ /a^ )〗^n)〗^(1/n) dx〗

Arc length of the curve :
Lx = ∫_0^x▒〖√( &1+〖f^ 〗^' 〖(x)〗^2 ) " " dx〗
〖f^ 〗^' 〖(x)〗^( )= 〖-b〗^ /a^ 〖(1-〖(x^ /a^ )〗^n)〗^((1/n)-1) 〖(x^ /a^ )〗^(n-1)
Lx = ∫_0^x▒〖√( &1+(〖〖b^ /a^ 〗^ )〗^2 〖(〖(1-〖(x^ /a^ )〗^n)〗^((1/n)-1) 〖(x^ /a^ )〗^(n-1))〗^2 ) " " dx〗

Hydraulic radius :
Rh = □(area/(wetted perimeter))= □(A/P)





Section 3: Square Roots
a ≥ b ≥ 0
√(a^2+b^(2 ) )= a + (5b^2)/(〖9a〗^ +〖3b〗^ ) …. eq.(1-3) √(a^2+b^(2 ) )
b
fig.(1-3) a

table (1-3)
a b (a^2+b^2)^.5 eq.(1-3)
1 1 1.4142136 1.416667
1 .9 1.3453624 1.346154
1 .8 1.2806248 1.280702
1 .7 1.2206556 1.220721
1 .6 1.1661904 1.166667
1 .5 1.118034 1.119048
1 .4 1.077033 1.078431
1 .3 1.0440307 1.045455
1 .2 1.0198039 1.020833
1 .1 1.0049876 1.005376
1 0 1 1






Section 4: Angles
π/2 ≥ θ ≥ 0
θ =(180/π)π/2 sin⁡θ (1-2/π cos⁡θ+0.423cos^2 θ-0.15 cos^3 θ )
………..eq.(1-4)
θ =(180/π)π/2 sin⁡θ (1-∑_(n=1)^(n=4)▒〖(-1)^n 〖(Rn)cos^n θ〗^ 〗 )
………..eq.(2-4)
R1=2/π ;R2= 0.47583;R3=0.29178 ;R4=0.09004
θ =(180/π)(sin⁡θ+ (1-cos⁡θ )^(4⁄3) ((1-cos⁡θ+0.1416 sin⁡θ)/(1-cos⁡θ+sin⁡θ ) ))
………..eq.(3-4)
table (1-4)
sin⁡θ cos⁡θ θ eq.(1-4) eq.(2-4) eq.(3-4)
0.000 1.000 0 0.000 0.000 0.000
0.342021 0.939692 20 20.033697 20.00944 19.96253
0.50000 0.866025 30 30.082325 30.00048 29.91656
0.707108 0.707105 45 45.07667 45.00019 44.89428
0.866027 0.5 0000 60 59.913662 60.00037 59.98335
0.984808 0.173645 80 79.895652 79.9783 80.11747
1.000 0.000 90 90.00021 90.00021 90.00004



Section 5: Perimeter & Arc Length of Ellipse

a



fig.(1-5)
a ≥ b ≥ 0
a ≥ b Ellipse ; a = b Circle
x^2/a^2 +y^2/b^2 =1 ; f(x) = y = b〖(1-〖(x^ /a^ )〗^2)〗^(1/2)
P=4(a+b- 〖0.8584ab〗^ /〖a+b〗^ ) ………..eq.(1-5)

Lx=x+(b-y)( (b-y+0.1416x)/(b-y+x)) 〖(〖b-y〗^ /b^ )〗^(1/3) ………..eq.(2-5)





table (1-5)

a b n P x θ y Lx L(a-x)
1 1 2 6.283 0.2 11.54 0.98 0.201212 1.37
1 1 2 6.283 0.5 30 0.866 0.522137 1.049
1 1 2 6.283 0.7 44.43 0.714 0.77352 0.797
1 1 2 6.283 1 90 0 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 30 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 1 90 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 30 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 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 30 0.26 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 90 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 30 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 90 0 1.021955 0


















Section 6:Area&Perimeter for (x^n/a^n +y^n/b^n =1)

L=a+b*(((2.5/(n+0.5))^(1/n))*b+a*(n-1)*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))/(b-y+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)+(n-1)*((1/(n+1))^(n-1))*(((x/a)^(2*n-1)))-(n-1)*((0.117)^(n-1))*(((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 90 0 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 90 0 1.119048 1.118034 0.25 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 90 0 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 90 0 1.454356 1.4478809 0.628 0 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 90 0 1.145651 1.1361172 0.314 0 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 90 0 1.010638 1.0061342 0.063 0 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 1.5 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 90 0 1.487859 1.4847092 0.688 0 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 90 0 1.165229 1.1575142 0.344 0 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 90 0 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 90 0 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 90 0 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 90 0 1.017157 1.0105798 0.073 0 0.07338752 0.073174
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 90 0 1.57075 1.5706394 0.785 0 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 0.5 2 1.214 4.855333 1 90 0 1.213833 1.2107791 0.393 0 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 90 0 1.021955 1.015305 0.079 0 0.07852991 0.0785398
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 90 0 1.691927 1.6860646 0.878 0 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 90 0 1.294104 1.2860697 0.439 0 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 1 90 0 1.037506 1.0313494 0.088 0 0.0877659 0.0883319
1 1 4 1.762 7.04689 0.2 11.54 1 0.2 0.2000009 0.919 0.72 0.1995097 0.199984
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 90 0 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 90 0 1.347985 1.3316356 0.46 0 0.45958962 0.463519
1 0.1 4 1.052 4.206541 0.2 11.54 0.1 0.2 0.2 0.092 0.072 0.01995097 0.0199984
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 90 0 1.051635 1.0434287 0.092 0 0.09191792 0.0927037
1 1 8 1.86 7.440225 0.2 11.54 1 0.2 0.2 0.971 0.771 0.19995143 0.2
1 1 8 1.86 7.440225 0.5 30 1 0.5 0.500001 0.971 0.471 0.50010841 0.499973
1 1 8 1.86 7.440225 0.7 44.43 0.993 0.7 0.700169 0.971 0.269 0.70183465 0.699432
1 1 8 1.86 7.440225 1 90 0 1.860056 1.8691588 0.971 0 0.97104234 0.978461
1 0.5 8 1.431 5.723451 0.2 11.54 0.5 0.2 0.2 0.486 0.386 0.09997572 0.1
1 0.5 8 1.431 5.723451 0.5 30 0.5 0.5 0.5000003 0.486 0.235 0.25005421 0.249986
1 0.5 8 1.431 5.723451 0.7 44.43 0.496 0.7 0.7000423 0.486 0.135 0.35091733 0.349716
1 0.5 8 1.431 5.723451 1 90 0 1.430863 1.4095861 0.486 0 0.48552117 0.48923
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 0.1 0.5 0.5 0.097 0.047 0.05001084 0.0499973
1 0.1 8 1.087 4.348056 0.7 44.43 0.099 0.7 0.7000017 0.097 0.027 0.07018347 0.0699432
1 0.1 8 1.087 4.348056 1 90 0 1.087014 1.0667961 0.097 0 0.09710423 0.0978461
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 1 90 0 1.916723 1.945223 0.993 0 0.99272763 0.996174
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
1 0.5 20 1.466 5.865887 1 90 0 1.466472 1.4614597 0.496 0 0.49636382 0.498087
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
1 0.1 20 1.105 4.420364 1 90 0 1.105091 1.0829743 0.099 0 0.09927276 0.0996174




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