Ellipse Ellipse Intersection

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tom-9000

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I have 2 ellipses:
Ellipse 1 is horizontal with the left focus located at 0, 0.
The length of the semi major axis = a1.
The length of the semi minor axis = b1.
The half focal separation = c1.

Ellipse 2 is vertical with the lower focus also located at 0, 0.
The length of the semi major axis = a2.
The length of the semi minor axis = b2.
The half focal separation = c2.

Variables: a1, b1, a2, and b2.
Constants: c1 and c2.
4c2 = 3c1

I need to find the location (X, Y) of the intersection point located in the first quadrant.

Thanks tom-9000
(If possible I would like to include an image)
 
tom-9000 said:
I have 2 ellipses:
Ellipse 1 is horizontal with the left focus located at 0, 0.
The length of the semi major axis = a1.
The length of the semi minor axis = b1.
The half focal separation = c1.

Ellipse 2 is vertical with the lower focus also located at 0, 0.
The length of the semi major axis = a2.
The length of the semi minor axis = b2.
The half focal separation = c2.

Variables: a1, b1, a2, and b2.
Constants: c1 and c2.
4c2 = 3c1

I need to find the location (X, Y) of the intersection point located in the first quadrant.

Thanks tom-9000
(If possible I would like to include an image)
Click to expand...

DUPLICATE POST

http://mathgoodies.com/forums/topic.asp?TOPIC_ID=33894

First draw a rough sketch of the problem - Locate the centers of the ellipses.

Next write the equations of the ellipses from the information given.

Then solve for the point of intersections in general you'll get four points of intersection.

For quick review - go to:

http://en.wikipedia.org/wiki/Ellipse

Please show us your work, indicating exactly where you are stuck - so that we know where to begin to help you.
 
Hello Subhotosh Khan

Thanks for your reply. I’ve worked on this problem for some time. I solved the ellipse circle intersection problem, but I can’t find the solution to ellipse ellipse intersection.

Your comments:
‘First draw a rough sketch of the problem’
The first illustration is a sketch of my problem.

‘Locate the centers of the ellipses’
As stated both ellipse 1 and ellipse 2 share a foci at (0, 0). Therefore, the center of ellipse 1 is at (c1, 0). And, the center of ellipse 2 is at (0, c2).

‘Next write the equations of the ellipses from the information given.’
The general formula for an ellipse is (X^2 / a^2) + (Y^2 / b^2) = 1.


’Then solve for the point of intersections’
Yes, this is the sticking point. The second illustration represents my best efforts to date.

‘in general you'll get four points of intersection.’
I think the specific parameters of my problem limit the number of intersections to two.

Thanks tom-9000[attachment=1:t5spnxer]EllipseEllipseIntersection.jpg[/attachment:t5spnxer][attachment=0:t5spnxer]Ellipse Equation.jpg[/attachment:t5spnxer]
 

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tom-9000 said:
Hello Subhotosh Khan

Thanks for your reply. I’ve worked on this problem for some time. I solved the ellipse circle intersection problem, but I can’t find the solution to ellipse ellipse intersection.

Yes, this is the sticking point. The second illustration represents my best efforts to date.

‘in general you'll get four points of intersection.’
I think the specific parameters of my problem limit the number of intersections to two.

Thanks tom-9000[attachment=1:19o78gn3]EllipseEllipseIntersection.jpg[/attachment:19o78gn3][attachment=0:19o78gn3]Ellipse Equation.jpg[/attachment:19o78gn3]
Click to expand...
Now you have an equation for X[sup:19o78gn3]2[/sup:19o78gn3]

It looks okay upto her.

Are you trying convince me to slog through the algebra for you??!!

No such luck - do it - it will build character (I had mine built long time ago)!!!
 
Hello Subhotosh Khan

Thanks but my character is just fine, it’s my algebra that is lacking. I don’t want you to ‘slog through’ but some idea of what the next step should be would be nice.

Thanks tom-9000
 
How about this:

Solve each ellipse equation in terms of Y. (Go one step further than you already have, taking the square root of each side. You'll get two equations from each ellipse: the "+" being the top half of the ellipse, and the "-" being the bottom half.)

Now set equations from ellipse 1 equal to equations from ellipse 2. (There are four possible combinations.)

The four combinations will result in anywhere from zero to four solutions, depending on the input parameters.
 
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