A Real Ellipse is a Planar Curve given by A Set of All Points on a Plane whose Sum of Distance from 2 Distinct Fixed Points on the Plane is Constant. The Fixed Points are the 2 Foci of the Ellipse.
An Imaginary Ellipse is a Mathematical Object having all Properties/Parameters Similar (if not same) to that of a Real Ellipse.
All Properties/Parameters that can be Mathematically Calculated for any Real Ellipse can also be Mathematically Calculated for any Imaginary Ellipse (for eg. the 2 Foci of the Ellipse).
However An Imaginary Ellipse unlike a Real Ellipse Cannot be Plotted or Traced on any Real Plane.
Henceforth All Properties/Parameters that are mentioned in this document for an Ellipse are applicable to both Real and Imaginary Ellipses.
The Distance between the 2 Foci of an Ellipse is denoted by letter \(\mathbf{F}\). Half the Distance between the 2 Foci is denoted by letter \(\mathbf{c}\).
The Constant Sum of Distance from 2 Foci is the Length of the Major Axis of the Ellipse and it is denoted by letter \(\mathbf{L}\). Half of Length of the Major Axis is called Semi-Major Axis and it is denoted by letter \(\mathbf{a}\).
The 2 End Points of the Major Axis are the 2 Vertices of the Ellipse.
The Mid Point of the 2 Vertices (or Mid Point of the 2 Foci) is the Center of the Ellipse.
The Line Segment Between 2 Points on Ellipse Passing through it's Center and Perpendicular to it's Major Axis is the Minor Axis of the Ellipse.
The 2 End Points of the Minor Axis are the 2 Co-Vertices of the Ellipse. The Distance Between the 2 Co-Vertices is the Length of Minor Axis of the Ellipse and it is denoted by letter \(\mathbf{l}\).
Half of Length of the Minor Axis is called Semi-Minor Axis and it is denoted by letter \(\mathbf{b}\).
Length of Major Axis (\(L)\), Length of Minor Axis (\(l)\) and the Distance between the 2 Foci (\(F)\) for any Ellipse are related as follows
\(F^2=L^2-l^2\)
Similar relation exists beween the Length of Semi-Major Axis (\(a)\), Length of Semi-Minor Axis (\(b)\) and Half the Distance between the 2 Foci (\(c)\) given as follows
\(c^2=a^2-b^2\)
The above 2 equations imply that For any Ellipse \(L > F\) (hence \(a>c\)).
The Angle that the Major Axis of Ellipse makes with the Poisitive Direction of \(X\)-Axis is the Angular Orientation of the Ellipse.
The 2 Lines Perpendicular to the Major Axis of the Ellipse at a Distance of \(\frac{L^2}{2F}\) (or \(\frac{a^2}{c}\)) from the Center of Ellipse on the either side of the Center of Ellipse are called the 2 Directrices of the Ellipse.
The Ratio of the Distance Between the 2 Foci To the Length of the Major Axis of Ellipse is called the Eccentricity of the Ellipse. It is denoted by letter \(\mathbf{e}\) and calculated as follows.
\(e=\frac{F}{L}=\frac{c}{a}\)
The Eccentricity of an Ellipse is Always \(\geq 0\) and \(< 1\).
The Eccentricity of an Ellipse can also be given as the Ratio of the Distance Between a Vertex and Center of Ellipse (which is the Length of Semi-Major Axis of Ellipse) To Distance Between the Directrix and Center of Ellipse. That is
The Eccentricity of an Ellipse can also be given as the Ratio of the Distance Between a Focus and a Point on Ellipse To Distance Between the Directrix (on the same side of Center as the Focus) and that Point on Ellipse.
The Line Segment Between 2 Points on Ellipse, Passing through its Focus, Parallel to it's Directrices and Perpendicular to it's Major Axis is called the Latus Rectum of Ellipse.
Each Ellipse has 2 Latus Recta, one Passing through one of the Focus and the other Passing through the other Focus.
The Length of each Latus Rectum is \(\frac{l^2}{L}\) (or \(\frac{2b^2}{a}\)).
In summary, following are the Parameters / Properties for any given Real or Imaginary Ellipse
Length of Major Axis/Semi-Major Axis
Length of Minor Axis/Semi-Minor Axis
Length of Latus Rectum
Distance/Half Distance Between 2 Foci
Distance of Directrices from the Center
Eccentricity
Angular Orientation
Coordinates of Center
Coordinates of 2 Vertices
Coordinates of 2 Co-Vertices
Coordinates of 2 Foci
Coordinates of Point of Intersection Between the 2 Directrices and Major Axis
Coordinates of Point of Intersection Between the 2 Latus Rectas and Ellipse
Equation of the Major Axis
Equation of the Minor Axis
Equation of the 2 Directrices
Equation of the 2 Latus Recta
In 2 Dimensions, Ellipse (both Real and Imaginary) can be of following 2 types
Axis Aligned Ellipse: The Axes of these Ellipse are Parallel/Perpendicular to the Coordinate Axes. Axis Aligned Ellipses can be of the following 2 subtypes
X-Major Ellipse: Ellipse having their Major Axis Parallel to \(X\)-Axis and Minor Axis Parallel to \(Y\)-Axis.
Y-Major Ellipse: Ellipse having their Major Axis Parallel to \(Y\)-Axis and Minor Axis Parallel to \(X\)-Axis.
Non-Axis Aligned Ellipse or Rotated Ellipse: The Axes of these Ellipse are Not Parallel or Perpendicular to Coordinate Axes.
For Axis Aligned Ellipse, the General Quadratic Equation in 2 Variables representing the Ellipse can be given in form of Standard Equations or Implicit Equations.
For Non-Axis Aligned Ellipse or Rotated Ellipse, the General Quadratic Equation in 2 Variables representing the Ellipse can be given in form of Implicit Equations only.