Equation or semi-axes of the ellipse

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Discussion Overview

The discussion revolves around the possibility of determining the equation or the semi-axes of an ellipse given two points on its arc, specifically focusing on the known y-coordinates and the difference of the x-coordinates. The context includes mathematical reasoning and exploration of the conditions under which such calculations can be made.

Discussion Character

  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant suggests that two points on the ellipse are insufficient to determine the ellipse's properties, as more information is typically required.
  • Another participant provides a specific example with two points and attempts to derive the semi-axes using equations derived from the ellipse's definition, questioning whether their approach contains a major flaw.
  • It is noted that the assumption of the ellipse being centered at the origin with axes parallel to the coordinate axes is critical for the validity of the calculations presented.
  • One participant reiterates the conditions of the ellipse and asks if it is possible to define the semi-axes given the known parameters.
  • Another response indicates that with the provided information, it leads to two equations with three unknowns, suggesting that a solution cannot be reached under these conditions.

Areas of Agreement / Disagreement

Participants generally agree that two points alone are insufficient to fully define the ellipse, but there is disagreement on the implications of the specific conditions provided and whether the semi-axes can be determined from the given information.

Contextual Notes

The discussion highlights limitations regarding the assumptions made about the ellipse's center and orientation, as well as the number of points required to define an ellipse accurately.

arkkis
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I have been struggling for a while now with this one.

Lets say I have two points on the arc of the ellipse. I know the y-coordinates and the difference of the x-coordinates. Is it possible two calculate the equation or the semiaxes of the ellipse where these points are located?

These are the information I can obtain. Actually the exact equation of the ellipse is not relevant. I should be able to calculate the ratio of the semi-axes.

EDIT: The ellipse is origo centered and the axes are parallel to x/y axes.

- - - -
Ellipse as a function of x:

x=sqrt[(a^2*b^2-a^2*y^2)/(b^2)]

So the difference of x-coordinates ie. deltax

deltax= x2-x1 =sqrt[(a^2*b^2-a^2*y2^2)/(b^2)] - sqrt[(a^2*b^2-a^2*y1^2)/(b^2)]

I guess somwhow I sould be able to eliminate a or b.
- - -
I also tried in a way where I know also the x-coordinates by solving a equation pair where i end up with:

a^4*(y2^2-y1^2)+a^2*(x2^2*y1^2-y2^2*x1^2)=0

I can't get it solved when just the x2-x1 is known.
- - -
Thanks!
 
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Two points are too few to determine much of anything about an ellipse.
 
mathman said:
Two points are too few to determine much of anything about an ellipse.

But if I have two points ie (1;4,71) and (2:3,73)

If I put these to the formula which is obtained from pair equation of an ellipse (x1;y1 and x2;y2):

(x1^2)/(a^2)+(y1^2)/(b^2)=1
(x2^2)/(a^2)+(y2^2)/(b^2)=1

Calculate b^2 and put to the latter:

a^4*(y2^2-y1^2)+a^2*(x2^2*y1^2-y2^2*x1^2)=0

a^2*[a^2*(y2^2-y1^2)+(x2^2*y1^2-y2^2*x1^2)]=0

a^2=2 OR a^2*(y2^2-y1^2)+(x2^2*y1^2-y2^2*x1^2)=0

The latter

a^2=[(y2^2*x1^2)-(y1^2*x2^2)]/(y2^2-y1^2)

a^2=[(3,73^2*1^2)-(4,71^2*2^2)]/(3,73^2-4,71^2)

a^2=9,0463 --> a=-3,01 or 3,01

Hence:
b^2=(a^2*y1^2)/(a^2-x1^2) = b^2=24,943
b = -4,99 or 4,99

Am I making a major flaw?
 
You are making an assumption that the ellipse is centered at the point (0,0) and the major / minor axes are not at angles to the x / y axes. If that's true, what you have done is valid. If the center is unknown, it takes 4 points to define an ellipse (assumming no axis angle), just as it takes 3 points to define a circle.
 
Yes. The ellipse is origo centered and the axes are parallel to x/y axes. In addition i know the y-coordinates of two points and difference of the x-coordinates. That should be it.

Is it possible to define the semi-axes?
 
arkkis said:
Yes. The ellipse is origo centered and the axes are parallel to x/y axes. In addition i know the y-coordinates of two points and difference of the x-coordinates. That should be it.

Is it possible to define the semi-axes?

The equation of the ellipse you are looking for is (x/a)2 + (y/b)2 = 1.
The semi-axes are a and b.

You can set up two equations in terms of (x1,y1) and (x2,y2). You are given x2-x1, y1, y2.
This gives you two equations in three unknowns a, b, x1. Can't be solved.
 

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