<< analytic_geometry

Ellipse

Basic Concepts

Definition

Ellipse is the plane curve that contains all the points that the sum of distance from two focal points all the same. That is, for two focal points \(F_{1}, F_{2}\), \(F_{1}P + F_{2}P = d > 0\). Usually we denoted the sum of distance \(d=2a\), and the distance between two focal, called focal distance or linear eccentricity, as \(2c\).

Analytical Expression

The equation of standard ellipse, which centered at the origin(that is, the two focal points are put at \(x\) axis symmetrically in \((c, 0), (-c, 0)\) respectively), computed as

\[ \sqrt{(x+c)^2 + y^2} + \sqrt{(x-c)^2 + y^2} = 2a \]

That is,

\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]

where we define \(b^2 = a^2 - c^2\).

Property

Obviously, this standard ellipse intersects with \(x, y\) axis as \((\pm a, 0), (0, \pm b)\) respectively, hence \(a, b\) are named as the semi-major and semi-minor axis.

Interaction

Interaction with Line

The length of chord that generates from standard form equation of ellipse and line can be expressed as:

\[ d = \frac{2\sqrt{a^2b^2(A^2+B^2)(a^2A^2+b^2B^2+C^2)}}{a^2A^2+b^2B^2} \]