Ellipse Major Axis Rotation

[rapids79]

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!

 

[quasar987]

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?

 

[rapids79]

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?

 

[quasar987]

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.

 

[blue_raver22]

Hello,

Thank you for your question. You are correct in your thinking that the rotation of the major axis does not change the way you determine if a point lies inside or outside of an ellipse. The equation for an ellipse is:

(x/a)^2 + (y/b)^2 = 1

Where a is the length of the semi-major axis and b is the length of the semi-minor axis. This equation does not change with the rotation of the major axis. However, the coordinates of the point may need to be adjusted to account for the rotation.

To find if a point (x,y) lies inside the rotated ellipse, you can use the same equation and simply substitute the rotated coordinates for x and y. For example, if the major axis is rotated 45 degrees, the new coordinates (x',y') can be found using the following equations:

x' = xcos(45) - ysin(45)
y' = xsin(45) + ycos(45)

Once you have the new coordinates, you can plug them into the equation for the ellipse and solve for x' and y'. If the resulting value is less than 1, then the point lies inside the rotated ellipse.

I hope this helps. Let me know if you have any other questions or need further clarification. Good luck with your work!