Point and ellipse (rotated) position test: algorithm

Question

How to test if a point P = [xp,yp] is inside/outside some rotated ellipse given by the centre C=[x,y], a, b, and phi ( angle of rotation)?

At this moment I am using the following solution: rotate ellipse and point by the angle -phi and then the common test for a position of the point and "non rotated" ellipse.

But there are a lot of tested points (thousands) and I find this solution as slow. Is there any direct and more efficient way to get a position of the rotated ellipse and point?

I do not need a code but the algorithm. Thanks for your help.

Answer 1

Another option is just to throw everything into the equation for a 2D rotated ellipse and see if the result is less than one.

So a point is inside the ellipse if the following inequality is true

Where (xp,yp) are the point coordinates and (x0, y0) is the center of the ellipse.

I implemented a small Mathematica program demonstrating that this indeed works:

Here it is in action:

And here is the code:

ellipse[x_, y_, a_, b_, \[Alpha]_, x0_: 0, y0_: 0] := 
     (((x - x0)*Cos[\[Alpha]] + (y - y0)*Sin[\[Alpha]])/a)^2
   + (((x - x0)*Sin[\[Alpha]] - (y - y0)*Cos[\[Alpha]])/b)^2;

Manipulate[
 RegionPlot[
  ellipse[x, y, a, b, \[Alpha] \[Degree], Sequence @@ pos] < 1, {x, -5, 5}, {y, -5, 5}, 
  PlotStyle -> If[ellipse[Sequence @@ p, a, b, \[Alpha] \[Degree], Sequence @@ pos] <= 1, Orange, LightBlue], 
  PlotPoints -> 25]
, {{a, 2}, 1, 5, Appearance -> "Labeled"}
, {{b, 4}, 2, 5, Appearance -> "Labeled"}
, {\[Alpha], 0, 180,  Appearance -> "Labeled"}
, {{p, {3, 1}}, Automatic, ControlType -> Locator}
, {{pos, {0, 0}}, Automatic, ControlType -> Locator}]