Ellipse’s: equation, foci, eccentricity

The equation of ellipse is usually written in one of 2 ways, as a parametric equation or as a Cartesian equation are as follows

P(h +a cos t ,k+ b sint), Where P is a point of the ellipse

or

where h and k are the amount the center of the ellipse is offset from the origin in the x and y directions and a and b the points on the x and y axis where the ellipse cuts.

An ellipse has to focal points (+/- ae) and can be defined as the loci of all the points P such that the sum of the distances from P to each focal point is constant.

The eccentricity of an ellipse e defines how sharp the ends of it are. e is always < 1 and can be calculated from the equation b² = a²(1-e²).

The directrices of an ellipse are x = ±a/e. These are lines such that the ratio of the distance from every point P on the ellipse from the directic and foci is the constant e, the eccentricity of the ellipse.

For example at the point (a,0) on the ellipse the distance to the directirc is

a/e - a = a(1/e - 1) = a(1-e)/e

and the distance from the ellipse to the foci is

a - ae= a(1-e)

the ration of these is

a(1-e)

a(1-e)/e

which equals e

and for any point p,

the distance to the directric is

a/e - acost = a(1/e - cost) = a(1-ecost)/e

when squared becomes

a²cost(e²cost-2e)+a²

e²

the distance to the foci is

√{(acost-ae)² + b²sin²t}

which with substitution for b given above becomes

a²cost(e²cost-2e)+a²

hence the ratio of the square of these 2 distances e² so the ratio of the distances themselves is e

Post by David Woodford, also avalible at www.breakingwave.co.nr