Ellipse

  link1      
  x-coordinate of Center O (h)    
  y-coordinate of Center O (i)    
  Semi-Major axis           a    
  Semi-Minor axis           b    
  Inclination of semi major axis on x-axis in degree: (θ)    
  Angle of radius vector w.r.t major axis &focus in degree (θ1)    
  Time in Earth days to revolve round the orbit    
         
  Put y-intercept of any st. line (y=mx + c)    
  Put slope of above st.line (m)    
  If point P (x1,y1) of ellipse subtends angle δ at center w.r.t x-axis (parametric angle t ) (in degree)    
  Find Out        
  parametric angle t=arctan[(b/a)*tanδ] (in degree)  (in radian)    
  x-coordinate of P (a*cost)    
  y-coordinate of P (b*sint)    
  Arc Length (δ=0 to δ ) -----    
  Area swept by radius vector through (δ=0 to δ )    
  area swept / arc length    
  area swept / arc length (experimental-ab/(a+b)    
  Area of ellipse    
  Perimeter(1)*    
  Perimeter(2)    
  Perimeter(3)    
  Area/Perimeter(1)    
  Area/Perimeter(2)    
  Area/Perimeter(3)    
  Eccentricity (e)    
  Ellipticity (b/a)    
  Linear Eccentricity √(a*a-b*b)    
  Focal length (from center to focus)    
  x-coordinate of Focus C1 (c1 )    
  y-coordinate of Focus C1 (d1)    
  x-coordinate of Focus C2 (c2)    
  y-coordinate of Focus C2 (d2)    
  Angle of inclination in radian (rk)    
  Angle of inclination of radius vector in radian(rk1)    
  Focus2focus length    
  Latus Rectum (2b2/a)    
  Value of radius vector from center:b / [1-(ecosθ1)2 ]1/2    
  Value of radius vector from focus:a(1-e2)/(1+ecosθ1)    
  Perihelion distance: a(1-e)    
  Aphelion distance : a(1+e)    
  Equ. of ellipse if θ is zero (x- )2 / ()   + (y- )2 / () =1    
  Equ. of ellipse if θ is  θ (1/)[x+y-]2 +(1/)[y-x-]2 =1    
  Equ. of ellipse if θ is  θ x2 + y2 + x+y

xy+=0

   
         
  Whether the st.line shall intersect the ellipse ?

c2 > a2m2 +b2 ;

   
  Equ. of orthoptic circle

[xcosθ+ysinθ-h]2+[-xsinθ+ycosθ-k)2   = a2+b2

[x+y-]2 +[y-x-]2 =    
         
  Area swept per hour (total area/orbital period)    
  Average Speed per hour (perimeter/orbital period)    
  speed at Apogee/Aphelion (2*area  swept per hr /apogee distance)    
  speed at perigee/perihelion(2*area swept per hr perigee distance)    
  Difference in Speed    
  difference of speed as % of average speed %    
  If one does not know a, b of the ellipse, fill information below &

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  Fill Apehelion distance (far)    
  Fill Perihelion distance (near)    
  Find a & b (a) ---- (b)    
  Find e    
         
    x2 + y2 + x  + y + =0 Ax2 +By2 +Cx+Dy+E=0    
    where A, B, C, D are constants    
  Does the Equation represent Ellipse ?

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  semi-major axis :    
  semi-minor axis:    
    eccentricity       :    
    x-coordinate of center:    
    y-coordinate of center:    
         
   x2 + y2 + x  + y+xy + =0 Ax2 +By2 +Cx+Dy+Exy+F=0    
  Does the Equation represent Ellipse ?

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  semi-major axis :    
  test diffe= semi-minor axis:    
    eccentricity       :    
    x-coordinate of center:    
    y-coordinate of center:    
    angle of major axis to x-axis:    
   E2 - 4AB --    
         
         
Equ. of ellipse whose center is origin (0,0) & major axis is along x-axis:

r1+r2=2a .......(1) by definition of ellipse

(c+x)2 + y2 =(r1)2  ......(2)

 

(c-x)2 + y2 =(r2)2  ......(3)

from (1) r1=2a-r2; Putting the value in (2),   (c+x)2 + y2=(2a-r2)2

Subtracting (3) from above, we get 4cx =4a2 - 4ar2 or

r2=( a2-cx)/a -------(4)

Putting this value in (3), we get x2(a2-c2) /a2 + y2=a2-c2 .Dividing both sides by a2-c2

 x2 / a2 +  y2 / b2=1  ----(5)  where   a2-c2= b2

If the center has co-ordinate (h,k), the equn. becomes

(x-h)2 / a2

 +  (y-k)2 / b2=1 ----(6)

If the ellipse is rotated by θ  in anti-clockwise direction around the centre, equn (5) becomes

(xcosθ+ysinθ)2 / a2 +  (-xsinθ+ycosθ)2 / b2=1  ----(7a)  where   a2-c2= b2

and equn (6) becomes

[xcosθ+ysinθ-h]2 / a2 +  [-xsinθ+ycosθ-k)2 / b2=1  ----(7b)

Distance of Radius Vector from Focus

x=rcosθ1 ; y=rsinθ1; Equn of ellipse (x+c)2 /a2  + y2/ b2  =1

(c+rcosθ1)2 / a2   +  r2sin2θ1 / b2 = 1 or

a2r2 + b2r2cos2θ1-a2r2cos2θ1+2b2crcosθ1 +b2c2-a2b2 =0 or

a2r2 -[(a2r2cos2θ1- b2r2cos2θ1)-2b2crcosθ1 -b2c2+a2b2 ]=0 or

a2r2 -[c2r2cos2θ1-2b2crcosθ1 -b2c2+a2b2 ]=0 or

a2r2 -[c2r2cos2θ1-2b2crcosθ1 +b4 ]=0 or

a2r2 -[crcosθ1 -b2 ]2=0  or

ar =± (crcosθ1 -b2 )     or

r=  b2 /(a+c*cosθ1) since c=ae

r=  b2 /a(1+e*cosθ1) or

r = ( b2 /a) /(1+ecosθ1 ) = [ (  a2- c2 )/a]/(1+ecosθ1 ) = a(1- e2 )/(1+ecosθ1 ) since c=ae;

Area : πab

*Perimeter(1): π(a+b)[1+3h/(10+√(4-3h))] where h=(a-b)2 / (a+b)2

The above is Ramanujan's formula for the ellipse.

Perimeter(2): 2π√( a2/2  + b2/2 ) .... that is 2pi times the r.m.s value of a,b

Parametric Equation:

x=h + acost  (dx/dt=x'=-asint)

y=k + bsint   (dy/dt=y'=bcost)

where (h,k) are co-ordinates of center.

Here t is not the angle subtended by the point(x.y)

tan t=(a/b)(y-k)/(x-h)=(a/b) tanδ  where δ is the angle subtended by the point (x,y) at the center

with respect to x-axis.

Arc Length : ∫ √(x'2 + y'2 ) dt = ∫ √(a2sin2t +b2cos2t)dt =∫ √(a2 -a2cos2t +b2cos2t)dt

=∫ a √(1 - e2cos2t)dt =∫ a[1+1/2(-e2cos2t) + (1/2!)(1/2)(-1/2)(-e2cos2t)2+..........]dt

since (1+x)n = 1+nx+(1/2!)n(n-1)x2 + (1/3!)n(n-1)(n-2)x3+......

=a[∫dt -(e2/2)∫ (cos2t)dt] -a∫(e4/8)(cos4tdt) +...........

=a[t- (e2/2) (t/2 + sin2t /4)] ------(2nd approximation)

=a[t- (e2/2) (t/2 + sin2t /4)- (e4/8)(sintcos3t + (3/8)sintcost+(3/8)t)] --3rd approximation

as ∫ cos2tdt = (t/2 + sin2t/4) +C

   ∫ cos4tdt = (3t/8 + 3sin2t/16+(1/4)sintcos3t) +C'

Area Swept :

area = (1/2) ∫ r2 dθ1

area = (ab/2) (E-esinE)  where

E=2arctan[tanδ/2  * √(1-e)/(1+e)]

or area=(ab/2)[ δ - arctan ( sin2δ *(b-a)/ ( b+a+{b-a}cos2δ ) ]

Conic Equation of ellipse :

Ax2 +By2 +Cx+Dy+E=0 represents an ellipse whose major/minor axis are parallel to rectangular x-axis or y-axis and in this case there is no xy term.

h=-C/(2 b2 ) ; k=-D/(2 a2)

A= b2 ; B=a2 ;C=-2 b2h; D=-2 a2k ; E=   b2h2+   a2k2 a2b2

--------------------------

Ax2 +By2 +Cx+Dy+Exy+F=0 represents an ellipse whose major/minor axis are not parallel to rectangular x-axis or y-axis and hence the ellipse is inclined to the axis unless xy coefficient is zero. On rearranging the general ellipse equation terms 7(b), we get the following-

 b2cos2θ+ a2sin2θ =A ........(8)

 b2sin2θ+ a2cos2θ =B ........(9)

 a2ksinθ - b2hcosθ =C/2 ........(10)

a2kcosθ + b2hsinθ =-D/2 ........(11)

 b2sin - a2sin2θ  = E

or sin2θ =E /(b2 -a2 ) ........(12)

 b2h2+a2k2-a2b2 =   F ..............(13)

(8) + (9) =b2+ a2 =A + B ........(14)

(8) - (9) = (cos2θ-sin2θ)(b2 -a2 ) =A-B 

or cos2θ  =(A-B) / (b2 -a2 ) ; ........(15)

(12) / (15) =tan 2θ = E / (A-B) .......(16)

Eliminating b2 from (8) & (9), we get

 a2= (Bcos2θ -Asin2θ )/ (cos2θ )............(17)

 b2= A+B-a2    .............(18)

From equation (10) & (11) by eliminating k, we get

h =(Dsinθ +Ccosθ )/(-2b2 ) ; ..........(19)

Similarly, by eliminating h, we get

k =(Csinθ - Dcosθ )/(2a2 ) ; ..........(20)

The problem arises in equation (16) as to whether

tan2θ = E / (A-B) or

tan2θ = E / (B-A)

This is a deep problem because we really do not know this and have to proceed with any one of the above assumptions to find out a & b.

If a > b , then our assumption is correct.else we have to take up the alternate assumption by discarding the first assumption.

How nature does it in physical cases is to be seen since the probability of taking up any assumption is 0.5.

It shall be observed that the co-efficient of xy do not play any role in deciding whether a curve will be a circle, parabola, ellipse or hyperbola. Appearance of co-efficient merely suggests that the curve is tilted in the x-y axis.

Equation (13) plays a vital role in deciding the shape of a curve.