Equation for Ellipse from a chord - no other parameters

[uno4short]

Equation for Ellipse from a chord -- no other parameters!

Homework Statement

Known conditions are:
End points of the chord intersect the major and minor axis.
Proximity to nearest parallel tangent.

Known (hypothetical) values are:
The length of the chord is 10.
The chord is 1.25 from the nearest parallel tangent.

Homework Equations

This is where I get hung. I cannot seem to find any that apply. So...

The goal -- "find equation" to solve any *one* of the following:

What are the (x,y) and (x',y') coordinate values for the foci on the ellipse?
What is the (x,y) coordinate values for the center of the ellipse relative to the chord?
What is the angle from the center of the ellipse to the center of that chord relative to the major or minor axis?
What is the angle of the chord relative to the major or minor axis?
What is the length of either the major or minor axis?

Basicly, I am looking to find any defining value anywhere on that ellipse so that I can make use of standard equations.

The Attempt at a Solution

This is one of many sample diagrams I've used to help analize the problem:
http://img124.imagevenue.com/img.php?image=29019_ChordEllipse_122_392lo.jpg

It is clear that, given the length of the chord and the distance to the parallel tangent, there can be only one solution (excluding the mirror image of the ellipse which for all practical purposes equates to "one and the same").

I have no values for a or b or C or anything that I can use to relate back to the standard equations. No line slopes to work with, so where can I go from here?

 

[zgozvrm]

This is one of many sample diagrams I've used to help analize the problem:
http://img124.imagevenue.com/img.php?image=29019_ChordEllipse_122_392lo.jpg

This is a dead link.

It is clear that, given the length of the chord and the distance to the parallel tangent, there can be only one solution (excluding the mirror image of the ellipse which for all practical purposes equates to "one and the same").

Not true. There are an infinite number of ellipses that fit that description. However, there are only 2 "standard" ones (with their axes parallel to the x- and y-axes).

(See attachment)

 

[uno4short]

Thanks for the reply zgozvrm.

This is a dead link.

I just clicked on it and it still works for me.

Not true. There are an infinite number of ellipses that fit that description. However, there are only 2 "standard" ones (with their axes parallel to the x- and y-axes).

Key words in your statement: "that fit that description".

Symantics aside. If you see car on the street and cock your head, that makes it a different car? Is that right? The relations of the chord on the ellipse remain the same no matter how you choose to turn it. Now, what do you have that's actually useful?

 

[zgozvrm]

I just clicked on it and it still works for me.

Yes, it work for me too (at home). It failed at work though (not blocked).

Symantics aside. If you see car on the street and cock your head, that makes it a different car? Is that right? The relations of the chord on the ellipse remain the same no matter how you choose to turn it.

Point taken. But if you're going to come up with an equation for this ellipse (and the coordinates of the foci), you'll need to narrow it down to just one ellipse.

My point is valid. There is not just one single answer. How can you come up with the coordinates of any point (foci, or the tangent point M in your picture) otherwise?

 

[uno4short]

Yes, it work for me too (at home). It failed at work though (not blocked).


Point taken. But if you're going to come up with an equation for this ellipse (and the coordinates of the foci), you'll need to narrow it down to just one ellipse.

My point is valid. There is not just one single answer. How can you come up with the coordinates of any point (foci, or the tangent point M in your picture) otherwise?

Indeed. Simi-valid at best. How does that statement apply to this question?:
What is the exact distance from the center of the chord to the center of the ellipse?
Any way you spin it, there is only one "correct" answer.

Seems we are beginning to hone in on the issue at hand. What is the right question to find the solution to the problem? I don't care how you spin it. I just need to find a way to relate to some "relative" value on the ellipse, so I can put that value into a standard equation and actually plot the curve.

 

[zgozvrm]

How does that statement apply to this question?:
What is the exact distance from the center of the chord to the center of the ellipse?
Any way you spin it, there is only one "correct" answer.

For THAT question, yes, you are correct. But, there are several questions being asked in the problem. You never stated that you were ONLY concerned with the distance from the center of the chord to the center of the ellipse.

There are 2 questions in your original post and 1 question in the picture you linked to that ask for coordinates. My point all along has been that you can't determine coordinates without more information being given (you can only find relative distances). Therefore, you can't come up with an equation describing the ellipse, nor can you graph it. You can merely draw one with the correct dimensions.

I just need to find a way to relate to some "relative" value on the ellipse, so I can put that value into a standard equation and actually plot the curve.

A "standard equation" for an ellipse describes a specific ellipse. For instance, the equation

$$\frac{(x-3)^2}{25} + \frac{(y+2)^2}{16} = 1$$

describes an ellipse that has the following characteristics:
center at (3, -2)
major axis 10 units long, parallel to the x-axis
minor axis 8 units long, parallel to the y-axis

The equation describes THAT ellipse only; If you "tilt" that ellipse by ANY amount, you need to find a different equation to describe it.

The problem you posed doesn't define where the center of the ellipse is located on the Cartesian plane, nor does it define the angle of the major axis in relation to the x-axis. So, there is no single answer.



And, by the way, although

$$\frac{(x-3)^2}{25} + \frac{(y+2)^2}{16} = 1$$

and

$$\frac{(x+6)^2}{16} + \frac{(y-5)^2}{25} = 1$$

describe ellipses with equal major and minor axes, in mathematics, they are NOT considered to be the same ellipse.


All this being said, if you are NOT interested in the questions concerning coordinates, then you shouldn't have posted them and this whole discussion would never have taken place.

 

[uno4short]

All this being said, if you are NOT interested in the questions concerning coordinates, then you shouldn't have posted them and this whole discussion would never have taken place.

Sorry. Didn't mean to frustrate anyone. I thought I had been rather clear about the problem when stating the goal when I said any *one* of the following:

The goal -- "find equation" to solve any *one* of the following:

What are the (x,y) and (x',y') coordinate values for the foci on the ellipse?
What is the (x,y) coordinate values for the center of the ellipse relative to the chord?
What is the angle from the center of the ellipse to the center of that chord relative to the major or minor axis?
What is the angle of the chord relative to the major or minor axis?
What is the length of either the major or minor axis?

The answer to that last option would completely independant of orientation and applicable to all ellipses having a chord length of 6 with proximity to the nearest parallel tangent being a value of 1, given that the chord terminates at the major and minor axis.

The equation used to present a *solution* was deliberately left to your option. Anyway, I'm getting closer to an answer so I will post it when I'm done. Thanks for your time.

 

[zgozvrm]

Here's how I'd start (using your diagram):

1) assume the ellipse is centered at (0, 0)
2) assume the major axis lies on the x-axis (therefore, the minor axis lies on the y-axis)
3) use the convention that "a" is half the length of the major axis and "b" is half the length of the minor axis. Therefore, the equation of the ellipse will be of the form

$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$

4) determine the equation for the line on which the chord lies (slope would therefore be -b/a)
5) using the distance of the parallel tangent to the chord, determine the equation of the parallel tangent (this is most likely where you're getting stuck)

a) using a known point on the chord, say (-a, 0) determine the equation of the circle whose center lies on that point and has a radius of h (in your picture, h=1)
b) determine the equation of the line perpendicular to the chord through the center of the circle.
c) find the intersection of that line with the circle (2 points)
d) using the "lower" point, find the equation of a line perpendicular to the line described in step "b" (this new line will be the tangent line to the ellipse, parallel to the chord)


From here, I'll leave it to you...

 

[uno4short]

5) using the distance of the parallel tangent to the chord, determine the equation of the parallel tangent (this is most likely where you're getting stuck)...

Trying to make use of that equation for an ellipse is where I got stuck. I had to abandon it to find some real values for a and b. Now that I've found a soultion to those values, I can go back and make use of the equation for an ellipse.

Here is my solution with a complete breakdown, also illustrated here:
http://img139.imagevenue.com/img.php?image=40251_Chord2Ellipse_122_404lo.jpg

Please keep in mind the original goal was essentially to determine any equation that would help me relate to the ellipse. This path just happened to lead me to values for a and b:

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

Starting with chord length k = 6
Distance to parallel tangent h = 1

Ratios for a circle inscribed in a square tells me:
s = ((s+u)*.707) <-- sine of 45 degrees

Segments of lines bisecting a rectangle are congruent so:
s = k/2


so, juggling for u (as seen in the illustration):
s = (s+u)*.707
or:
s = .707*s + .707*u
or:
s - .707*s = .707*u
or:
u = s/.707 - s
and substituting for s:
u = (k/2)/.707 - (k/2)
or:
u = (6/2)/.707 - (6/2)
or:
u = 1.2426406871192851464050661726291


Now I have h and u, so:

o = sqrt(u^2-h^2)
or:
o = sqrt(1.2426^2 - 1^2)
or:
o = 0.73766921942310235015417348295202


Given the above, I can now determine the "relative" angle from point M to the center of the ellipse and relative distance between the two (k/2+u).

Now having determined the dimentions of that single triangle, I can easily find "relative" values for any other component of the ellipse. Thats all I needed to get going.

atan(h/o) = relative angle from point M to Center of ellipse.
b = 2*sqrt( s^2 - (cos(atan(h/o)/2)*s)^2 ) ---> 2.70456
a = sqrt( k^2-b^2 ) ---> 5.35587

All values above relative to the chord itself. Rotate your coordinate system as you like.

More than one way to skin a cat. I have my solution now so I probably won't be following this thread for much longer. Feel free to share though if you have any thoughts. Others may follow.

Just want to close by saying thanks for your time zgozvrm, and all your efforts to help stimulate a final resolution.

 

[zgozvrm]

Ratios for a circle inscribed in a square tells me:
s = ((s+u)*.707) <-- sine of 45 degrees

Segments of lines bisecting a rectangle are congruent so:
s = k/2

But we're not necessarily dealing with a circle, so you can't depend on those relationships.
Remember, all circles are ellipses, but not all ellipses are circles.

 

[uno4short]

But we're not necessarily dealing with a circle, so you can't depend on those relationships.
Remember, all circles are ellipses, but not all ellipses are circles.

Perspective. For *all* circles inscribed in a square (or vice versa), these relationships remain constant reguardless how you skew it, squash it, stretch it or rotate it. Its a fact. Please observe...

http://img270.imagevenue.com/img.php?image=75779_Sine45_122_140lo.jpg

All ellipses are circles and all circles are the same. Only perspectives change. It would help you to keep that in mind. Just as any 4 sided parallelogram is a square. Thats why we study math -- to learn about the relatonships as persectives change.

 

[zgozvrm]

All ellipses are circles and all circles are the same. Only perspectives change. It would help you to keep that in mind. Just as any 4 sided parallelogram is a square. Thats why we study math -- to learn about the relatonships as persectives change.

First of all, only certain cases of ellipses are circles; those with equal major and minor axes. In other words, circles are ellipses in which every point on the ellipse is an equal distance from the center of the ellipse.

Secondly, not every 4-sided parallelogram is a square. A square, by definition has 4 equal sides and 4 right angles.

Lastly, the picture you linked to is quite obviously incorrect. In the first (upper-left) drawing, there is a circle inscribed in a square. Although you didn't state this, the lengths of the sides of the square (and, therefore, the diameter of the circle) are equal to 1 unit. The diagonal of such a square is 1.414 (the square root of 2), so the distance from the center of the square (or circle) to any of the 4 vertices of the square will be equal to 0.707. You correctly indicate this in red and pink.

Now, if you cut the red diagonal line off at the point where it intersects the circle, it becomes shorter. In fact, it's length would be 0.500 (the radius of the circle). But, you show the radius in black and gray as being 0.707 units long. That is not possible.

Also, you can easily see that in the 3rd drawing the pink diagonal is MUCH shorter than the red diagonal, yet you show them both as being equal to 0.707 units.

 

[uno4short]

... the picture you linked to is quite obviously incorrect.

The only thing that is obvious is that you decided you could draw concrete conclusions about the relationships illustrated in the diagram without actually printing it out and measuring each component. If you had then you would have observed the .707 relationships are accurate in each case, including the lower illustration, which incidentally is identical to the two above seen only from a different *perspective*. Go ahead. Dare to take a step outside of the textbook into the real world, print it out and measure each component. If you come back with any more arguments then I (and eveyone else that has been following this conversation) will know you are afraid to leave the textbook and that you come here only to assert an aire of superior knowledge. Of course if that were the case, you might have *actually* presented a working set of equations to resolve the delimma that confronted me to begin with before I did, and you didn't. Already thinking up your next argument? Of course you are. It would be a lot easier to blow more hot air than it would be to print out the illustration and measure. Please spare me the grief and just go back into your textbook world... thank you very much.

 

[zgozvrm]

You obviously don't want any help, or just want to argue points that you cannot prove.

I've checked my work both mathematically ("in the textbook," as you say) and using tools like AutoCAD to back up my statements.

Believe what you want, but if you're not willing to listen to reason, then don't ask for help.

I'll spare you any more "grief" ... I'm done (my time can be better spent help those that understand fundamental properties of mathematics and geometry).


Good luck.