Titled, 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:

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

 

[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

 

[CellCoree]

I understand the importance of having a reliable and efficient equation for your research. After reviewing the information you provided, I believe the formula you have suggested is correct. It takes into account all the necessary parameters, such as the coordinates of the center, major and minor axis lengths, and angle of tilt. It is also a well-known formula for an ellipse that has been translated and rotated. However, I would recommend double-checking the formula and testing it on a few sample points to ensure its accuracy before using it for your entire dataset of 2,500 points.

Additionally, I would suggest looking into any available software or programming tools that can automate this process for you. This would save you time and effort in manually calculating the position of each point within the ellipse. Overall, it seems like you have a solid approach to solving this problem and I wish you the best of luck with your research.