Titled, Translated Ellipse

  • Thread starter Sleeker
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  • #1
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I need to find whether or not a point is within an ellipse. The problem is that the ellipse is tilted at an angle and not at the origin. I've tried Googling everywhere and can't find a good equation for what I need. Does anybody know the formula for an ellipse that includes:

1. Coordinates of the ellipse's center
2. Length of major axis (diameter or radius)
3. Length of minor axis (diameter or radius)
4. Angle the ellipse is tilted relative to x or y axis (doesn't matter which, I can figure it out from there).

I'm doing astronomical research, and I'm trying to locate points within galaxies which are shaped like ellipses. The four things I listed are the things I am given.

Edit: I know I can translate and rotate my ellipse, but I would really like just one formula since I need to do this approximately 2,500 times for my astronomical research.

Another edit: Maybe this?

[tex](\frac{x cos\theta+y sin\theta - x_c}{a})^2 + (\frac{x sin\theta-y cos\theta - y_c}{b})^2 = 1[/tex]

a = major axis (radial)
b = minor axis (radial)
[tex]x_c[/tex] = x coordinate of center
[tex]y_c[/tex] = y coordinate of center
[tex]\theta[/tex] = Angle of tilt from x-axis

I kind of just mixed and matched formulas until I think I incorporated everything. Is it right?
 
Last edited:

Answers and Replies

  • #2
Stephen Tashi
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It may be simplest to compute a change of coordinates that tranforms to a coordinate system where the ellipse is not tilted. Then apply the change of coordinates to the points in question and solve the problem in the simpler setting.
 
  • #3
LCKurtz
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Try this:

[tex]\frac{((x-x_c)\cos\theta + (y-y_c)\sin\theta)^2}{a^2}+
\frac{((x-x_c)\sin\theta - (y-y_c)\cos\theta)^2}{b^2}=1[/tex]
 
  • #4
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Hm, yeah, that makes more sense with incorporating the fact that it's off-center with the formula from this website:

http://www.maa.org/joma/Volume8/Kalman/General.html [Broken]
 
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