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general ellipse equation

x2
--
a2		 + y2
--
b2		 = 1
<=> y2
--
b2		 = 1 - x2
--
a2
<=> y2 = b2 * ( 1 - x2
--
a2		 ) => y = sqrt [ b2 * ( 1 - x2
--
a2 		)]
3 measured points P1(x1 | y1) ; P2(x2 | y2) ; P3(x3 | y3)

used equation in this case (half ellipse moved in y-direction):
y = - sqrt [ b2 * ( 1 - x2
--
a2 		)] + y0

1. possible solution


y = -sqrt( b2/a2 * (a2 - x2) ) + y0
y = -sqrt( b2/a2 ) * sqrt (a2 - x2) + y0

set z := -sqrt( b2/a2 )
y = z * sqrt (a2 - x2) + y0

system of equations:
equation I:
y1 = z * sqrt (a2 - x12) + y0
equ. II:
y2 = z * sqrt (a2 - x22) + y0
compute equation I-II:
y1-y2 = z * ( sqrt (a2 - x12) - sqrt (a2 - x22) )
<=> z = (y1-y2) / ( sqrt (a2 - x12) - sqrt (a2 - x22) )
so one gets:
y2-y3 = z * ( sqrt (a2 - x22) - sqrt (a2 - x32) )
<=> z = (y2-y3) / ( sqrt (a2 - x22) - sqrt (a2 - x32) )
equalize z terms:
(y1-y2) / ( sqrt (a2 - x12) - sqrt (a2 - x22) )
= (y2-y3) / ( sqrt (a2 - x22) - sqrt (a2 - x32) )
get the reciproke values on both sides:
( sqrt (a2 - x12) - sqrt (a2 - x22) ) / (y1-y2)
= ( sqrt (a2 - x22) - sqrt (a2 - x32) ) / (y2-y3)
set y12 := y1-y2 and y23 := y2-y3
y23 * ( sqrt (a2 - x12) - sqrt (a2 - x22) )
= y12* ( sqrt (a2 - x22) - sqrt (a2 - x32) )
square of both sides:
y232 * ( (a2 - x12) - 2*sqrt ((a2 - x12)*(a2 - x22)) + (a2 - x22))
= y122 * ( (a2 - x22) - 2*sqrt ((a2 - x22)*(a2 - x32)) + (a2 - x32))

2. solution

set z: = (1 - x2 / a2)
y = -sqrt( b2 * z ) + y0
...

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(this page has been updated on August-29-2003)