2.20. Eccentricity Of Conic Sections

Description of the conics in vector algebra terms made use a quantity called e.

Definition: Conics

Given a directrix line L and a focus point F not on L, a conic section is a set of points X satisfying

        XF     =ed( X,L )                (2.20)

where e is a positive real number called the eccentricity and d( X,L )  is the distance from X to L.  The conic is an ellipse if e<1 , a parabola if e=1 , and a hyperbola if e>1 .

Alternative Form

Now, given a point P on L, we can write

        d( X,L )= | ( XP )N | N   =    | ( XP ) N ˆ |

where N ˆ  is the unit normal vector of L.  Hence, (2.20) can be written as

        XF =e| ( XP ) N ˆ |                   (2.21)

This can be simplified by setting P=F+d N ˆ , where d is the distance between F and L [see Fig.2.12].  Thus,

          XF =e| ( XF ) N ˆ d |          (2.22)