How can I identify whether a point is within an ellipse which is not orthogonal in orientation

Question

I'm looking for an equation which will tell me whether or not a point in two-dimensional space, is located within an ellipse of known dimensions and orientation, and that is not orthogonal in nature.

Answer 1

I would first find the foci of the ellipse. If half the major axis is $a$ and half the minor axis is $b$, then the distance from the center to each focus is $c$ where $a^2-b^2=c^2$. From that and the center itself and angular orientation of the major axis (both given) you can get the foci. Then add up the distances from your point P to both foci.

Sum < major axis means P is inside ellipse

Sum = major axis means P is on ellipse

Sum > major axis means P is outside ellipse.