How to Determine if a Point Lies Inside a Tilted and Translated Ellipse?

[Sleeker]

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?

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

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

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

 

[Stephen Tashi]

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.

 

[LCKurtz]

Try this:

\frac{((x-x_c)\cos\theta + (y-y_c)\sin\theta)^2}{a^2}+<br /> \frac{((x-x_c)\sin\theta - (y-y_c)\cos\theta)^2}{b^2}=1

 

[Sleeker]

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