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Math Help - The foci of an ellipse - a bit unusual

  1. #1
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    Smile The foci of an ellipse - a bit unusual

    Hey guys,

    I'm trying to find the foci of an ellipse. If you know the major and minor axis, this is easy.

    I'm my case, I know the major axis and the radius of the ellipse.

    Is there an easy way then to find the foci?

    Thanks,
    M

    Edit: just realised there may not be a unique ellipse? Any opinions?
    Last edited by kingofinismac; August 18th 2009 at 02:51 AM.
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  2. #2
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    Quote Originally Posted by kingofinismac View Post
    I'm trying to find the foci of an ellipse. If you know the major and minor axis, this is easy.

    I'm my case, I know the major axis and the radius of the ellipse.

    Is there an easy way then to find the foci?
    What do you mean by the radius of an ellipse?
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  3. #3
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    I mean I know the sum of r1 and r2, the two distances from the foci that determine the shape of the ellipse

    I suppose I can find the foci by getting the point along the major axis that is (r1+r2) away from the end of the major axis?
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  4. #4
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    Quote Originally Posted by kingofinismac View Post
    I mean I know the sum of r1 and r2, the two distances from the foci that determine the shape of the ellipse

    I suppose I can find the foci by getting the point along the major axis that is (r1+r2) away from the end of the major axis?
    I'm still not understanding this. If r1 and r2 are the distances from a focus to the two ends of the major axis, then their sum will be the length of the major axis. So if you already know the length of the major axis, then knowing r1+r2 won't tell you anything new.
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    Sorry, let me clarify.



    I know the length of the major axis, 2a.

    I also know the value of r1 +r2. Therefore, if I know the intersection point of the major axis and the ellipse, and r1+r2, I can get the foci, can't I?
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    Quote Originally Posted by kingofinismac View Post
    Sorry, let me clarify.



    I know the length of the major axis, 2a.

    I also know the value of r1 +r2. Therefore, if I know the intersection point of the major axis and the ellipse, and r1+r2, I can get the foci, can't I?
    r1 + r2 = 2a. So as already mentioned, you don't have enough information to uniquely define the ellipse.
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  7. #7
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    r1 + r2 = 2a. So as already mentioned, you don't have enough information to uniquely define the ellipse.
    Agreed, but I think there may be another solution, specific to the specific problem I'm working on.

    I'm working on a kind of sonar problem. I'm transmitting a signal from one loudspeaker to a microphone. The signal is transmitted outwards, meets an obstacle, is reflected and recieved at a microphone, as shown in the diagram below:



    I know the total distance from the loudspeaker to the obstacle and back to the microphone. The signal can't go behind the loudspeaker (to the right) or to the left of the microphone, so this makes it a special type of ellipse. I think the obstacles location is on an ellipse where the foci and the end points of the major axis overlap?!

    I'm not sure, I just think there should be a unique ellipse for this problem?

    Thanks
    M
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  8. #8
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    Quote Originally Posted by kingofinismac View Post
    I know the total distance from the loudspeaker to the obstacle and back to the microphone. The signal can't go behind the loudspeaker (to the right) or to the left of the microphone, so this makes it a special type of ellipse. I think the obstacles location is on an ellipse where the foci and the end points of the major axis overlap?!

    I'm not sure, I just think there should be a unique ellipse for this problem?
    If the foci are at the ends of the major axis then the minor axis will be of length zero, and the "ellipse" will just consist of a line segment along the major axis. That obviously isn't the case here. The fact that the signal can't go beyond the microphone/loudspeaker just tells you that the reflector can only consist of part of an ellipse. But it doesn't tell you anything about the shape or eccentricity of the ellipse.

    Do you know the distance between the microphone and the loudspeaker? If you do, then that will determine the shape of the ellipse.
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    Yes, the two pieces of information I know are the distances between the loudspeaker and microphone and r1+ r1.
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  10. #10
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    Quote Originally Posted by kingofinismac View Post
    Yes, the two pieces of information I know are the distances between the loudspeaker and microphone and r1+ r1.
    If an ellipse has semimajor axis a and eccentricity e, then the distance between the foci is 2ae, and the length of the reflected path from one focus to the other is r1+r2=2a. In this case, the loudspeaker and microphone are situated at the foci. If the distance between them is d, then the eccentricity of the ellipse is given by e=(r1+r2)/d. The semiminor axis b is then given by b = \sqrt{a^2(1-e^2)}, and that determines the shape of the ellipse.
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  11. #11
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    Okay, great.

    Thank you very much for the help guys.

    It's very much appreciated!

    M
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