The Ellipse

ellipse

A curved planar figure, the locus of all points which have the same total distance from two fixed points called the foci.

Geometrically, an ellipse can be defined as:
Given two fixed points F₁ and F₂, an ellipse is the figure formed from all points P for which PF₁ + PF₂ = 2a.

The two fixed points are called the foci (plural of focus).

Analytically, an ellipse can be defined as:
Given two constants a and b, an ellipse is the locus of all points (x, y) for which

x2   y2     ----- + ----- = 1 a2   b2

• The geometric definition and the analytic definition are equivalent, and will produce the same figure for foci (-c, 0) and (c, 0), where a² = b² + c²

Axes

• The line segment from (-a, 0) to (a, 0), is called the major axis of the ellipse.
• The line segment from (0, -b) to (0, b), is called the minor axis of the ellipse.
• The line segment from (0, 0) to (a, 0), is called the semimajor axis of the ellipse.
• The line segment from (0, 0) to (0, b), is called the semiminor axis of the ellipse.
• These terms are also used to refer to the lengths of these line segments.

Intercepts

• An ellipse (in the standard form) intercepts the x axis at (a, 0) and (-a, 0).
• An ellipse (in the standard form) intercepts the y axis at (0, b) and (0, -b).

Excentricity

• The excentricity e of a ellipse is
  

  		c
  e	=	----
  		a

  where a² = b² + c²
• For all ellipses, e < 1;

Directrices

• Two directrices (plural of directrix), D₁ and D₂, can be defined for an ellipse, where
  D₁ is the line x = - a / e, and
  D₂ is the line x = a / e,
• For point P on the ellipse, and each directrix,
  PF₁ = e ⋅ PD₁, and
  PF₂ = e ⋅ PD₂

• The equation
  Ax² + Bxy + Cy² + Dx + Ey + F = 0
  will describe an ellipse if the discriminate,
  B² - 4AC,
  is less than 0.

• A ellipse can also be defined parametrically:
  Given two constants a and b and a parameter θ, an ellipse is the locus of all points (x, y) for which
  x = a cos θ, and
  y = b sin θ

• A ellipse can also be defined in polar coordinates
  Given a constant e less than 1, an ellipse is the locus of all points (ρ, θ) for which

  		1
  ρ	=	----------------
  		1 + e cos θ