How Does Rotating an Ellipse's Major Axis Affect Its Equation?

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Discussion Overview

The discussion revolves around the effects of rotating the major axis of an ellipse on its equation and the implications for determining whether a point lies inside the ellipse. Participants explore the mathematical methods and transformations involved in this scenario.

Discussion Character

  • Exploratory
  • Mathematical reasoning

Main Points Raised

  • One participant questions how the equation of an ellipse changes when its major axis is rotated, specifically at 45 degrees, and whether this affects the method for checking if a point lies inside the ellipse.
  • Another participant suggests that a point in the rotated ellipse corresponds to a point in the non-rotated ellipse when the coordinates are transformed by a rotation of -45 degrees.
  • A further inquiry is made about the mathematical methods available for checking if a point lies within the rotated ellipse, including a reference to a tangential method.
  • It is proposed that to determine if a point lies within the rotated ellipse, one must first rotate the point back by -45 degrees and then check its position relative to the non-rotated ellipse.
  • A rotation matrix is mentioned as a tool for transforming the coordinates of points, with a specific angle of -pi/4 suggested for the rotation.

Areas of Agreement / Disagreement

Participants appear to agree on the concept that rotating the point can help determine its position relative to the non-rotated ellipse, but the exact methods and implications of the rotation remain under discussion without a clear consensus.

Contextual Notes

There is an implicit assumption that the rotation of the ellipse does not change its fundamental properties, but this is not explicitly confirmed. The discussion also lacks detailed mathematical derivations or examples of the rotation matrix application.

Who May Find This Useful

This discussion may be useful for individuals interested in geometry, computer graphics, or software development related to mathematical modeling and transformations.

rapids79
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Hi..I have a basic question regarding the equation of an ellipse. Let's say I ahve an ellipse with major and minor axes 2a and 2b respectively. Now, to check whether a point lies inside this ellipse, its fairly simple...I can just use the standars ellipse equation for that. Now, if my major axis is rotates to..lets say 45 degrees, how does the equation of the ellipse vary and how do I find then whether a certain point lies within the ellipse?? Logically, rotation of the major axis must not change the way I look for a point to be located inside/outside an ellipse. except that let's say once the rotation angle is >= 90 degrees your major axis becomes minor an vice-versa. Is my thinking correct?? or does the equation and method vary? I'd really appreciate it if someone can throw me some pointers...thanks..



PS: Before any1 asks, this is not a homework problem..am a workign professional..cheers!
 
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A point is part of the 45 degree-rotated ellipse <==> the -45 degree-rotated point is part of the nonrotated ellipse. Does that help?

What are you a working professional of, might I inquire?
 
Hi..Thanks for the reply..am just starting out in software...need to implement a code that does that for a project...so, anyway...if I have a point A...how would I check if it lies within the rotated ellipse...is there any mathematical method like the tangential method that does that?
 
According to the idea of post #2, you need to show that the point A, when rotated by -45 degree, is in the non rotated ellipse. Do you see that?

If so, then all you need is an equation for the coordinates of a point rotated by -45 degrees. The function R^2-->R^2 which rotates every point by an angle O is linear and its matrix is called a rotation matrix. Check out wikipedia. Applying this matrix on the left to a column vector (x y) will give you a column vector whose entries are the x and y coordinates of the rotated point. Use O=-pi/4 in this formula to get the result for the ellipse rotated by 45 degrees.
 

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