Jesús de la Peña Hernández
Under these circumstances, any line through P 1 intersecting its polar and the ellipse produces a
set of four points whose harmonic cross ratio is:
(P ́P ́ ́P 1 P 2 ) = -1 (1)
Let ́s see the meaning of this expression. Point P 1 of Fig. 4 has slid down in Fig. 5 to the X-axis
so (1) will lie on it (look the four rounded points).
Let x 1 be the abscissa of the 1st point P ́ in (1), x 2 of P ́ ́, x 3 of P 1 and x 4 of P 2. The cross ratio
(1) has the value:
: 1
2 41 4
2 3(^13) =−
−
−
−
−
x x
x x
x x
x x
(2)
In (2) we know all the abscissas but x 3 to which we assign the value x. Substituting in (2) the
known values, we have:
: =− 1
−
− −
−
− −
a c
a c
a x
a x
From which we obtain the distance x between the center O of the ellipse and the directrix d:
c
a
x
2
= (3)
Now we can show the folding process to construct the directrix (Fig. 6):
1- F ́ (after step 3, Point 13.3.1) → Small axis of the ellipse ; O → O
Result: E.
2- EF ́ → EG ; E → E.
Result: EH, hence C.
3- AC → CI ; C → C.
Result: d
Note that x in (3) is greater than the abscissa of the point symmetric to F about the el-
lipse ́s vertex. These abscissas ́ difference is:
()
()
c
a c
a a c
c
a
2 2
−
− + − =
obviously greater than zero.
(3) can also be proved geometrically just looking at the right ∆EF ́C in which OC = x.
In it we have:
OE^2 =OF ́×OC ; a^2 =cx
Here it is another property of the directrix d (Fig. 7): the ratio of distances from any
point P(x,y) on the ellipse to the directrix (PE) and to its corresponding focus (r = PF), equals
the ellipse ́s eccentricity (e = c / a).
O
F ́
6
A C
E
cp d
H
I
G