For an elliptical spot, I want to calculate the average diameter (or radius). As you can see in the picture, the diameter depends on the coordinate system chosen. If you rotate the coordinate system, then the diameter obtained are different. In the picture these diameters are [a,b] and [x,y] --> depending on the coordinate system chosen. Now if I take average for these two set of diameters, which way will be a better choice? The arithmetic mean changes, while rotating the coordinate system. So, a+b is not equal to x+y. Does the RMS diameter remains constant with respect to the coordinate system? Is, a^2 + b^2 = x^2 + y^2 ?
If (x0, y0) is the centre and (x, y) is a point on the ellipse, in any rectilinear coordinate system, you get the radial separation as:
r = √( (x-x0)2 + (y-y0)2 )
When you then average r over the full ellipse, it should not vary with the choice of coordinate system. Subtracting the coordinates of the centre deals with arbitrary translations and the equation for the radial separation deals with any arbitrary rotation.
What is your intention for finding the mean radius of an ellipse? RMS is a measure of deviation from a mean. A circle would have an RMS=0 about the mean radius, of course, and the more extreme the elliptical shape, the higher the RMS about the mean radius would be. Probably the easiest way to get a mean radius of an ellipse is to find the radius of a circle which would have the same area as the ellipse. pi*a*b = pi*r^2 therefore the mean radius r = sqrt(a*b).
Michael Linnolt : I do not think you read my question and attached image carefully.
I was not asking for the definition of geometric mean - which you stated.
The question is simple:
Q) What is the difference in RMS and AM if you rotate the coordinate axis? Will RMS or AM remain same with this rotation?
Also, your idea about RMS is wrong. Please read the Wikipedia article on RMS (https://en.wikipedia.org/wiki/Root_mean_square)
I asked the question again and this time received the correct answer. Check Below:
https://www.researchgate.net/post/Is_RMS_a_better_way_to_take_average_than_arithmatic_mean
Prasenjit Roy My answer is perfectly correct as I stated it. RMS can have several meanings, depending on the application. I gave you the definition of a std dev about the mean. Your question as initially stated is quite unclear as to what you are trying to do! The mean diameter of an ellipse, is best found by setting the area equal to a circle, since that would be the typical usage for example, optical imaging, where you want to find the equivalent spot diameter. So, it would be most helpful to specify what your intended application or goal is, when stating a question!
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