This document provides a summary of new simpler equations for calculating properties of Lame curves, which include ellipses, hyperellipses, superellipses, and astroids. The author derives approximate equations to find square roots, angles, areas, perimeters, and arc lengths for these curves. The equations are presented as alternatives to the traditional approaches using integrals, and are intended to make calculations simpler. A history of solutions for ellipse properties is also provided.

1. New Simpler Equations for Properties of Lame Curve (Hypoellipse ,Ellipse,
Superellipse and Astroid Curves)
Author: Maher Ezzeddin Aldaher ; B sc.C. E. ; MosulUniv. 1987;
C. E. Municipality of Irbid West;Jordan
E- mail : maher_daher2000@yahoo.com;
Mobile no.: +962795133967,+962776733564
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 Lame curve, Hypoellipse ,Ellipse , (Hypercircle, Supercircle), Squircle
,(Hyperellipse, Superellipse, Rectellipse, Hyperoval ,Superoval , Peit Hein Superellipse) ,(Asteroid, a special case -
four cusped hypocycloid , tetracuspid, cubocycloid, paracycle), in 2D plan, other related properties such as Hydraulic
radius found easily.
Key words: Lame curve, Perimeter, Area, Hydraulic radius.
Symbols:
a : major radius on x axis A : area in 1st quadrant Ax : partial area
At : total area in 4 quadrants b : minor radius on y axis L : arc length in 1st quadrant
Lx : partial arc length n : power varies from 1 to ∞ N : power varies from 0 to 1
P : perimeter in 4 quadrants Pw : wetted perimeter Rh : hydraulic radius

2. 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,also for general Astroid
area and perimeter lastly in 2012 ,the problem is to execute results from unsolvable integrals.
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, collected
scientific material can be found in selected references which mentioned at the end of the research.
It's an applicable research for different sectors such as applied mathematics, statistics , physics,
hydraulics, heat transfer, aerodynamics , mechatronics, antennas, botany, live beings, medical images,
computer vision, machineries, technicians, designer engineers, etc.. .Equations can be used for
Superellipse fitting and 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.
Section 2: Basic Concepts
The discovery of conic sections is credited to Menaechmus in Ancient Greece around the years 360-
350 B.C. These curves were later investigated by Euclid, Archimedes and Apollonius, the Great
Geometer. Conic sections were nearly forgotten for 12 centuries until Johannes Kepler (1571-1630).
Astroid (4-Cusped Hypocloid) discovered first by Olaus Roemer 1674
The general Cartesian notation of the form comes from the French mathematician Gabriel
Lamé (1795–1870) who generalized the equation for the ellipse (Lamé curve).
Piet Hein used the Lamé curve in many designs. He used n=2.5 found by trial and error and names it
the superellipse.
𝑎 ≥ 𝑏 ≥ 0 , ∞ ≥ n ≥ 1 , 1 > N > 0
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

3. n>2 , a>b Lame curve (Hyperellipse, Superellipse, Rectellipse, Hyperoval, Superoval)
n=2.5 , a>b Peit Hein Superellipse ; n=∞ , a=b Square ; n=∞ , a>b Rectangle
0<N<1 , a>b General Astroid
N=
2
3
, a=b special case of Astroid - four cusped hypocycloid , tetracuspid, cubocycloid, paracycle
Superellipse curve canbe writteninPolarform:
AlsoinParametriceqs.:
x(θ) = ±a cos2/n
(θ)
y(θ) = ±b sin2/n
(θ)
(0 ≤ θ < π/2)
fig.(1-2) Several shapes of
𝑥 𝑛
𝑎 𝑛 +
𝑦 𝑛
𝑏 𝑛 = 1

4. -100
-80
-60
-40
-20
0
20
40
60
80
100
-100 -80 -60 -40 -20 0 20 40 60 80 100
-100
-80
-60
-40
-20
0
20
40
60
80
100
-100 -80 -60 -40 -20 0 20 40 60 80 100
-100
-80
-60
-40
-20
0
20
40
60
80
100
-100 -80 -60 -40 -20 0 20 40 60 80 100
-100
-80
-60
-40
-20
0
20
40
60
80
100
-100 -80 -60 -40 -20 0 20 40 60 80 100
-100
-80
-60
-40
-20
0
20
40
60
80
100
-100 -80 -60 -40 -20 0 20 40 60 80 100
-100
-80
-60
-40
-20
0
20
40
60
80
100
-100 -80 -60 -40 -20 0 20 40 60 80 100
-100
-80
-60
-40
-20
0
20
40
60
80
100
-100 -80 -60 -40 -20 0 20 40 60 80 100
-100
-80
-60
-40
-20
0
20
40
60
80
100
-100 -80 -60 -40 -20 0 20 40 60 80 100
fig.(2-2) General Asteroids related to Superellipse,Hypoellipse
Area under the curve :
Ax = ∫ 𝑓( 𝑥) 𝑑𝑥
𝑥
0
= ∫ b(1 − (
𝑥
𝑎
) 𝑛
)1/𝑛
𝑑𝑥
𝑥
0
Arc length of the curve :
Lx = ∫ √1 + 𝑓 ′
(𝑥)2 𝑑𝑥
𝑥
0
𝑓 ′
(𝑥) =
−𝑏
𝑎
(1 − (
𝑥
𝑎
) 𝑛
)
(
1
𝑛
)−1
(
𝑥
𝑎
) 𝑛−1
Lx = ∫ √1 + (
𝑏
𝑎
)2((1− (
𝑥
𝑎
) 𝑛)
(
1
𝑛
)−1
(
𝑥
𝑎
) 𝑛−1)2 𝑑𝑥
𝑥
0
Hydraulic radius :
Rh =
𝐚𝐫𝐞𝐚
𝒘𝒆𝒕𝒕𝒆𝒅 𝒑𝒆𝒓𝒊𝒎𝒆𝒕𝒆𝒓
= 𝑨
𝑷𝒘
Special cases can be solved to find area and arc length by direct integration such as:
Area & perimeter of diamond , circle , 4- cusped hypocycloid(astroid of N=2/3) , Area of ellipse.
Perimeter of ellipse can be solved using complete and incomplete elliptic integrals of second kind –
E(π/2,m) , E(Ф,m)

5. Area & perimeter of other shapes solved by applying generalized binomial theorem with Gamma - Г
& Beta - В special functions.
Section 3: History of related solutions
Complete Perimeter of Ellipse
1- Jonannes Kepler, 1609.
2- Takakazu Seki Kowa 1680.
3- Colin Maclurin 1742.
4- Leonard Euler 1773.
5- Landen 1775.
6 – Lagrange 1784.
7- Sipos 1792.
8- James Ivory 1796.
9- Thomas Muir 1883.
10- Charles Hermite (1882-1901).
11- Giuseppe Peano 1889.
12- Henry Pade (1-6) 1892.
13- Lindner, 1904.
14- Srinivasa Aiyangar Ramanujan (1&2)
1914.
15- Gauss-Kummer 1917.
16- Ralph G. Hudson 1917.
17- E.H. Lockh wood (1&2) 1932.
18- Necat Tasdelen 1959.
19- Ernest S. Semler (1-3) 1975.
20- ALmkvist, 1978.
21- Bronstein & Semendyayev, 1985.
22- Jocobsen & Waadhand, 1985.
23- J.M. Broweins & Broweins, 1987.
24- Borowski & Browein, 1991.
25-Maher Ezzeddin Aldaher 1992.
26- David F. Rivera (1&2), 1997.
27- Roger Maertens 2000.
28- David W. Cantrell 2001.
29- David W. Cantrell 2004.
30- David W. Cantrell-Ramanujan 2004.
31- Lu Chee Ket, 2004.
32- David F.Rivera (1-3) 2004.
33- Richardo Bartolomeu – G.P. Michon, 2004.
34- Stansilav. Sy’kora, 2005.
35- Salter, 2005.
36- David W. Cantrell 2006
37-Jean-Pierre Michon, 2007.
38- Paul Abbott, 2009.
39- Khaled Hassan Abed, 2009.
40- Shahram Zafary (1&2), 2009.
41- Benoitcn, 2011.
42- Ara. Kalaimaran, 2012.
43- Quadratic Formula. (More than author)

6. Incomplete Perimeter of Ellipse
1- Jonannes Kepler, 1609.
2- Takakazu Seki Kowa 1680.
3- Leonard Euler 1773.
4- Landen 1775.
5 – Lagrange 1784.
6- Sipos 1792.
7- A.M. Legendre 1794.
8- Thomas Muir 1883.
9- Giuseppe Peano 1889.
10- Henry Pade (1-6) 1892.
11- Lindner, 1904.
12 -Srinivasa Aiyangar Ramanujan(1&2) 1914.
13- E.H. Lock wood 1932.
14- Necat Tasdelen 1959.
16- Ernest S. Semler (1-3) 1975.
17- ALmkvist 1978.
18-Maher Ezzeddin Aldaher 1992.
19- Roger Maertens 2000.
20- David W. Cantrell 2001.
21- David W. Cantrell 2004.
22- David W. Cantrell-Ramanujan 2004.
23- David F. Rivera 2004.
24- Ricardo Bartolomeu G.P. Michon 2004
25- Stansilav. Sy’kora, 2005.
26- Salter 2005.
27- David. W. Cantrell, 2006.
28- Paul Abbott, 2009.
For Superellipse &Astroid Area
1- George B. Mc Clellan Zerr 1894.
2- Michael Brozinsky & Murray S. Klamkin
1995.
3- Maher Ezzeddin Aldaher 1996,2004&2012.
4-K. Kono 2011.
For Superellipse&Astroid Perimeter
1- Maher Ezzeddin Aldaher 1996&2012
2 - Valeri Astanoff 2009
3 - Necat Tasdelen 2010
4 - K. Kono 2011
Section 4: Square Roots
√𝒂 𝟐 + 𝒃 𝟐a ≥ b ≥ 0
√𝒂 𝟐 + 𝒃 𝟐 = a +
𝟓𝒃 𝟐
𝟗𝒂 +𝟑𝒃
…. eq.(1-4) b
a
fig.(1-4) Perpendicular Triangle

7. table (1-4) Comparing results of square roots
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 5: Angles
𝜋
2
≥ θ ≥ 0
θ = (
180
𝜋
)
𝜋
2
sin θ (1 −
2
𝜋
cosθ + 0.423cos2
θ − 0.15 cos3
θ ) … …. . eq. (1 − 5)
θ = (
180
𝜋
)
𝜋
2
sin θ (1 − ∑ (−1) 𝑛( 𝑅𝑛)cos 𝑛
θ𝑛=4
𝑛=1 ) ………..eq.(2 - 5)
𝑅1 =
2
𝜋
; 𝑅2 = 0.47583; 𝑅3 = 0.29178 ; 𝑅4 = 0.09004
θ = (
180
𝜋
)(sinθ + (1 − cosθ )
4
3⁄
(
1−cos θ+0.1416 sin θ
1−cos θ+sin θ
)) ………..eq.(3-5)
table (1-4) Comparing results for angles
𝐬𝐢𝐧 𝛉 𝐜𝐨𝐬 𝛉 θ eq.(1-5) eq.(2-5) eq.(3-5)
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 6: Perimeter & Arc Length of Ellipse
fig.(1-6) Ellipse curve

8. a ≥ b ≥ 0
a ≥ b Ellipse ; a = b Circle
𝑥2
𝑎2 +
𝑦2
𝑏2 = 1 ; f(x) = y = b(1 − (
𝑥
𝑎
)2
)1/2
P=4(a+b-
0.8584𝑎𝑏
𝑎+𝑏
) ………..eq.(1-6)
Lx=x+(b-y)(
b−y+0.1416x
b−y+x
) (
𝑏−𝑦
𝑏
)1/3
………..eq.(2-65)
table (1-6) Results of arc length and perimeter of Ellipse
Section 7:Area&Perimeter for Superellipse,Hypoellipse
a≥b , ∞≥ n ≥ 1 ,
𝒙 𝒏
𝒂 𝒏 +
𝒚 𝒏
𝒃 𝒏 = 𝟏
𝐋 = 𝐚 + 𝐛 (
𝐛(
𝟐.𝟓
𝐧+𝟎.𝟓
)
𝟏
𝐧
+
𝟎.𝟓𝟔𝟔( 𝐧−𝟏) 𝐚
𝐧 𝟐
𝐛+
𝟒.𝟓𝐚
𝐧 𝟐+𝟎.𝟓
) ………..eq.(1-7)
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

9. 𝐋𝐱 = 𝐱 + ( 𝐛 − 𝐲)
(
( 𝐛 − 𝐲)(
𝟐. 𝟓
𝐧 + 𝟎. 𝟓
)
𝟏
𝐧
+
𝟎. 𝟓𝟔𝟔( 𝐧− 𝟏) 𝐱
𝐧 𝟐
𝐛 − 𝐲 + (
𝟒. 𝟓𝐚
𝐧 𝟐 + 𝟎. 𝟓
) 𝐱
)
(
𝐛 − 𝐲
𝐛
)
(
𝐧−𝟏
𝟑
)
… … …. . eq. (2 − 𝟕)
P=L*4 ………..eq.(3-7)
L(a-x)=L-Lx ………..eq.(4-7)
A=a b(0.5) 𝑛−1.52
………..eq.(5-76)
At=A*4 ………..eq.(6-7)
A(a-x) = ay((𝟎.𝟓) 𝐧−𝟏.𝟓𝟐
-
𝐧 𝟒
𝟏+𝐧 𝟒 (
𝐱
𝐚
)+(n-1)(
𝟏
𝐧+𝟏
) 𝐧−𝟏
(
𝐱
𝐚
) 𝟐𝐧−𝟏
− (n-1)𝟎. 𝟏𝟏𝟕 𝐧−𝟏
(
𝐱
𝐚
) 𝟐𝐧−𝟏
) …..eq.(7-7)
Ax = A – A(a-x) ...…………eq.(8-7)
Other forms
A=a b(1 −
𝟏
2𝑛1.22
) ………..eq.(9-7)
By using Binomial nested series for expanding f(x)
Ax =b(x-
𝐚
𝐧(𝐧+𝟏)
(
𝐱
𝐚
) 𝐧+𝟏
+
𝐚( 𝟏−𝐧)
𝟐𝐧 𝟐( 𝟐𝐧+𝟏)
(
𝐱
𝐚
) 𝟐𝐧+𝟏
−
𝐚( 𝟏−𝐧)( 𝟏−𝟐𝐧)
𝟔𝐧 𝟑( 𝟑𝐧+𝟏)
(
𝐱
𝐚
) 𝟑𝐧+𝟏
+
𝐚( 𝟏−𝐧)( 𝟏−𝟐𝐧)( 𝟏−𝟑𝐧)
𝟐𝟒𝐧 𝟒( 𝟒𝐧+𝟏)
(
𝐱
𝐚
) 𝟒𝐧+𝟏
− … ………)
……..eq.(10-7)
Ax =bx+ab∑ (−1) 𝑚𝑚=∞
𝑚=1
∏ ( 𝟏−( 𝐦−𝟏) 𝐧)𝐦
𝐦=𝟏
𝐦!𝑛 𝑚( 𝐦𝐧+𝟏)
(
𝐱
𝐚
) 𝐦𝐧+𝟏
…… . eq. (11− 7)
A=ab(1-
𝟏
𝐧(𝐧+𝟏)
+
( 𝟏−𝐧)
𝟐𝐧 𝟐( 𝟐𝐧+𝟏)
−
( 𝟏−𝐧)( 𝟏−𝟐𝐧)
𝟔𝐧 𝟑( 𝟑𝐧+𝟏)
+
( 𝟏−𝐧)( 𝟏−𝟐𝐧)( 𝟏−𝟑𝐧)
𝟐𝟒𝐧 𝟒( 𝟒𝐧+𝟏)
- ……) …. …eq.(12-7)

10. table (1-7) Comparing results of arc length, perimeter, area , for Lame curve (Superellipse,Hypoellipse)
a b n L P maher x θ y Lx maher Lx Excel At A(a-x) A(x) maher A(x) mathm
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 0.7 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 0.7 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 0.7 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 0.7 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 0.7 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 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 0.7 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 0.7 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 0.7 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 0.7 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 0.7 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 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 0.7 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 0.7 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

11. Section 8:Area&Perimeter for General Asteroid
0<N<1 , a>b General Asteroid
N=
2
3
, a=b special case of Asteroid - four cusped hypocycloid , tetracuspid, cubocycloid,
paracycle
(
𝐱
𝐚
)
𝐍
+ (
𝐲
𝐛
)
𝐍
= 𝟏 ………..eq.(1-8)
fig.(1-8) Asteroid N=2/3 ,a=b ; Asteroid N=0.4 a>b
To find perimeter and area of general Asteroid ,first convert the power N to another similar
superellipse curve of power n.
For 0< N <1 , values of n estimated as:
1.000.840.70.6670.60.5620.50.3750.330.290.198N
1.001.21.491.581.822.02.44.05.37.020.0n
as)8-eq.(3) and8-eq.(2sthe executed relationFind graph program trial version 2.411by using
follows:
n=
( 𝟏.𝟖𝟖𝟐𝟔𝟐−𝟒.𝟏𝟐𝟕𝟓𝟕𝐍+𝟓.𝟑𝟏𝟒𝟓𝑵 𝟐−𝟐.𝟐𝟏𝟎𝟕𝟓𝑵 𝟑)
𝑵−𝟎.𝟏𝟑𝟔𝟏𝟐𝟐
………..eq.(2-8)
N=
( 𝟎.𝟓𝟎𝟏𝟒𝟖𝟗+𝟎.𝟐𝟓𝟕𝟓𝟒𝟕𝐧−𝟎.𝟎𝟎𝟖𝟒𝟏𝟗𝟒𝒏 𝟐+𝟎.𝟎𝟎𝟎𝟐𝟎𝟑𝟐𝟐𝟒𝒏 𝟑)
𝒏−𝟎.𝟐𝟒𝟗𝟐𝟏
………..eq.(3-8)
x coordinate of astroid= a - x coordinate of superellipse
y coordinate of astroid= b - y coordinate of superellipse

12. fig.(2-8)
L asteroid = L superellipse ………..eq.(4-8)
𝐋 = 𝐚 + 𝐛 (
𝐛(
𝟐.𝟓
𝐧+.𝟓
)
𝟏
𝐧
+
𝟎.𝟓𝟔𝟔( 𝐧−𝟏) 𝐚
𝐧 𝟐
𝐛+
𝟒.𝟓𝐚
𝐧 𝟐+𝟎.𝟓
) ………..eq.(5-8)
Lx asteroid = L(a-x) superellipse ………..eq.(6-8)
a-x of asteroid = x1 ………..eq.(7-8)
𝐋(𝐱𝟏) = 𝐱𝟏 + ( 𝐛 − 𝐲𝟏)
(
( 𝐛 − 𝐲𝟏)(
𝟐. 𝟓
𝐧+. 𝟓
)
𝟏
𝐧
+
𝟎. 𝟓𝟔𝟔( 𝐧− 𝟏) 𝐱𝟏
𝐧 𝟐
𝐛 − 𝐲𝟏 + (
𝟒. 𝟓𝐚
𝐧 𝟐 + 𝟎. 𝟓
) 𝐱𝟏
)
(
𝐛 − 𝐲𝟏
𝐛
)
(
𝐧−𝟏
𝟑
)
………..eq.(8-8)
Lx asteroid = L - Lx1 ………..eq.(9-8)
P=L*4 ………..eq.(10-8)
A asteroid = a b – a b(0.5) 𝑛−1.52
………..eq.(11-8)
At asteroid = A asteroid *4 ……..eq.(12-8)
A(x1) = ay𝟏((𝟎. 𝟓) 𝒏−𝟏.𝟓𝟐
-
𝒏 𝟒
𝟏+𝒏 𝟒 (
𝒙𝟏
𝒂
)+(n-1)(
𝟏
𝒏+𝟏
) 𝒏−𝟏
(
𝒙𝟏
𝒂
) 𝟐𝒏−𝟏
− (n-1)𝟎. 𝟏𝟏𝟕 𝒏−𝟏
(
𝒙𝟏
𝒂
) 𝟐𝒏−𝟏
)
……eq.(13-8)
Ax asteroid= bx - A(x1) ………..eq.(14-8)

13. table (1-8) Results of arc length, perimeter, area , for Asteroids
a b N n L P x x1 y y1 Lx1 Lx x1/a A At A(x1) A(x)
1 1 0.789 1.3 1.454 5.817 0.5 0.5 0.335 0.665 0.606 0.848 0.5 0.372 1.488 0.188 0.312
1 1 0.789 1.3 1.454 5.817 1 1E-05 3E-07 1 1E-05 1.454 1E-05 0.372 1.488 0.628 0.372
1 0.8 0.789 1.3 1.317 5.267 0.5 0.5 0.268 0.532 0.573 0.744 0.5 0.298 1.19 0.151 0.249
1 0.8 0.789 1.3 1.317 5.267 1 1E-05 3E-07 0.8 1E-05 1.317 1E-05 0.298 1.19 0.502 0.298
1 0.5 0.789 1.3 1.146 4.583 0.5 0.5 0.167 0.333 0.533 0.612 0.5 0.186 0.744 0.094 0.156
1 0.5 0.789 1.3 1.146 4.583 1 1E-05 2E-07 0.5 1E-05 1.146 1E-05 0.186 0.744 0.314 0.186
1 0.3 0.789 1.3 1.062 4.248 0.5 0.5 0.1 0.2 0.514 0.548 0.5 0.112 0.446 0.056 0.094
1 0.3 0.789 1.3 1.062 4.248 1 1E-05 1E-07 0.3 1E-05 1.062 1E-05 0.112 0.446 0.188 0.112
1 0.1 0.789 1.3 1.011 4.043 0.5 0.5 0.033 0.067 0.503 0.508 0.5 0.037 0.149 0.019 0.031
1 0.1 0.789 1.3 1.011 4.043 1 1E-05 3E-08 0.1 1E-05 1.011 1E-05 0.037 0.149 0.063 0.037
1 1 0.667 1.582 1.502 6.008 0.5 0.5 0.225 0.775 0.555 0.947 0.5 0.292 1.168 0.244 0.256
1 1 0.667 1.582 1.502 6.008 1 1E-05 2E-08 1 1E-05 1.502 1E-05 0.292 1.168 0.708 0.292
1 0.8 0.667 1.582 1.358 5.433 0.5 0.5 0.18 0.62 0.539 0.819 0.5 0.234 0.934 0.195 0.205
1 0.8 0.667 1.582 1.358 5.433 1 1E-05 1E-08 0.8 1E-05 1.358 1E-05 0.234 0.934 0.566 0.234
1 0.5 0.667 1.582 1.173 4.693 0.5 0.5 0.113 0.387 0.519 0.655 0.5 0.146 0.584 0.122 0.128
1 0.5 0.667 1.582 1.173 4.693 1 1E-05 9E-09 0.5 1E-05 1.173 1E-05 0.146 0.584 0.354 0.146
1 0.3 0.667 1.582 1.078 4.312 0.5 0.5 0.068 0.232 0.509 0.569 0.5 0.088 0.35 0.073 0.077
1 0.3 0.667 1.582 1.078 4.312 1 1E-05 5E-09 0.3 1E-05 1.078 1E-05 0.088 0.35 0.212 0.088
1 0.1 0.667 1.582 1.015 4.061 0.5 0.5 0.023 0.077 0.502 0.513 0.5 0.029 0.117 0.024 0.026
1 0.1 0.667 1.582 1.015 4.061 1 1E-05 2E-09 0.1 1E-05 1.015 1E-05 0.029 0.117 0.071 0.029
1 1 0.44 3 1.692 6.768 0.5 0.5 0.048 0.952 0.502 1.19 0.5 0.122 0.489 0.368 0.132
1 1 0.44 3 1.692 6.768 1 1E-05 6E-13 1 1E-05 1.692 1E-05 0.122 0.489 0.878 0.122
1 0.8 0.44 3 1.528 6.113 0.5 0.5 0.038 0.762 0.502 1.026 0.5 0.098 0.391 0.295 0.105
1 0.8 0.44 3 1.528 6.113 1 1E-05 5E-13 0.8 1E-05 1.528 1E-05 0.098 0.391 0.702 0.098
1 0.5 0.44 3 1.294 5.176 0.5 0.5 0.024 0.476 0.501 0.793 0.5 0.061 0.245 0.184 0.066
1 0.5 0.44 3 1.294 5.176 1 1E-05 3E-13 0.5 1E-05 1.294 1E-05 0.061 0.245 0.439 0.061
1 0.3 0.44 3 1.153 4.611 0.2 0.8 0.064 0.236 0.808 0.345 0.8 0.037 0.147 0.028 0.032
1 0.3 0.44 3 1.153 4.611 0.5 0.5 0.014 0.286 0.501 0.652 0.5 0.037 0.147 0.111 0.039
1 0.3 0.44 3 1.153 4.611 1 1E-05 2E-13 0.3 1E-05 1.153 1E-05 0.037 0.147 0.263 0.037
1 0.1 0.44 3 1.038 4.15 0.5 0.5 0.005 0.095 0.5 0.537 0.5 0.012 0.049 0.037 0.013
1 0.1 0.44 3 1.038 4.15 1 1E-05 6E-14 0.1 1E-05 1.037 1E-05 0.012 0.049 0.088 0.012
1 1 0.378 4 1.762 7.047 0.5 0.5 0.02 0.98 0.5 1.262 0.5 0.081 0.323 0.413 0.087
1 1 0.378 4 1.762 7.047 1 1E-05 4E-15 1 1E-05 1.762 1E-05 0.081 0.323 0.919 0.081
1 0.8 0.378 4 1.594 6.377 0.5 0.5 0.016 0.784 0.5 1.094 0.5 0.065 0.259 0.33 0.07
1 0.8 0.378 4 1.594 6.377 1 1E-05 3E-15 0.8 1E-05 1.594 1E-05 0.065 0.259 0.735 0.065
1 0.5 0.378 4 1.348 5.392 0.5 0.5 0.01 0.49 0.5 0.848 0.5 0.04 0.162 0.206 0.044
1 0.5 0.378 4 1.348 5.392 1 1E-05 2E-15 0.5 1E-05 1.348 1E-05 0.04 0.162 0.46 0.04
1 0.3 0.378 4 1.191 4.765 0.5 0.5 0.006 0.294 0.5 0.691 0.5 0.024 0.097 0.124 0.026
1 0.3 0.378 4 1.191 4.765 1 1E-05 1E-15 0.3 1E-05 1.191 1E-05 0.024 0.097 0.276 0.024
1 0.1 0.378 4 1.052 4.207 0.5 0.5 0.002 0.098 0.5 0.552 0.5 0.008 0.032 0.041 0.009
1 0.1 0.378 4 1.052 4.207 1 1E-05 4E-16 0.1 1E-05 1.052 1E-05 0.008 0.032 0.092 0.008
1 1 0.199 20 1.917 7.667 0.5 0.5 3E-05 1 0.5 1.417 0.5 0.007 0.029 0.493 0.007
1 1 0.199 20 1.917 7.667 1 1E-05 2E-29 1 1E-05 1.917 1E-05 0.007 0.029 0.993 0.007
1 0.8 0.199 20 1.737 6.947 0.5 0.5 3E-05 0.8 0.5 1.237 0.5 0.006 0.023 0.394 0.006
1 0.8 0.199 20 1.737 6.947 1 1E-05 2E-29 0.8 1E-05 1.737 1E-05 0.006 0.023 0.794 0.006
1 0.5 0.199 20 1.466 5.866 0.5 0.5 2E-05 0.5 0.5 0.966 0.5 0.004 0.015 0.246 0.004
1 0.5 0.199 20 1.466 5.866 1 1E-05 1E-29 0.5 1E-05 1.466 1E-05 0.004 0.015 0.496 0.004
1 0.3 0.199 20 1.286 5.145 0.5 0.5 1E-05 0.3 0.5 0.786 0.5 0.002 0.009 0.148 0.002
1 0.3 0.199 20 1.286 5.145 1 1E-05 7E-30 0.3 1E-05 1.286 1E-05 0.002 0.009 0.298 0.002
1 0.1 0.199 20 1.105 4.42 0.5 0.5 3E-06 0.1 0.5 0.605 0.5 7E-04 0.003 0.049 7E-04
1 0.1 0.199 20 1.105 4.42 1 1E-05 2E-30 0.1 1E-05 1.105 1E-05 7E-04 0.003 0.099 7E-04

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