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
‖
X−F
‖ =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
)=

(
X−P
)⋅N

‖ N ‖
= 
(
X−P
)⋅
N
ˆ

where
N
ˆ
is the unit normal vector of L.
Hence, (2.20) can be written as
‖
X−F
‖=e
(
X−P
)⋅
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,
‖
X−F
‖=e
(
X−F
)⋅
N
ˆ
−d

(2.22)