Eccentricity and directrix

The conics can be defined in terms of one of their foci, of a straight line called directrix and a number called eccentricity. The locus of the points such that the ratio of the distance to the focus and the distance to the directrix is equal to the eccentricity, is a conic curve.  If the eccentricity is less than 1.0 then it is an ellipse, if it is equal to 1.0 it is a parabola and if is is greater than 1.0 then it is a branch of a hyperbola.

This applet shows the general equation of a conic section in terms of the focus F, the directrix D and the angle t between D and the vertical.

The nippe Descartes has two special forms of equations which represent conic curves.  The first is based on the focust, the directrix and the eccentricity and consists of the explicit formula e=PF/PD, that is to say:

e=sqrt((x-Fx)^2+(y-Fy)^2)/(d+(x-Fx)*cos(t)+(y-Fy)*sen(t))

There is another special form for conic sections based on the focus, the directrix and a point of the curve.

The following applet uses this other form which consists of writing the relationship PF=e*PD explicitely:

e=sqrt((x-Fx)^2+(y-Fy)^2)/(d-((x-Fx)*(Dx-Fx)+(y-Fy)*(Dy-Fy))/d)

where

d=sqrt((Dx-Fx)^2+(Dy-Fy)^2)

and the eccentricity is calculated using the point P:

e=sqrt((Px-Fx)^2+(Py-Fy)^2)/(d-((Px-Fx)*(Dx-Fx)+(Py-Fy)*(Dy-Fy))/d)

Author: José Luis Abreu León

	Ministerio de Educación, Cultura y Deporte. Año 2000