Jesús de la Peña Hernández

13.3.3 ELLIPSE INSCRIBED WITHIN A RECTANGLE

Let ́s recall first the generation of a conic by means of projective line pencils. Fig. 1

shows the pencils V and V ́ inscribed in an ellipse in such a manner that the corresponding

lines intercept in points lying on the ellipse.

In a figure like that, both line pencils are projective, i.e., when intercepted by any lines

like a and a ́, produce two sets of four points (rounded in the drawing) with the same value of

their respective cross ratios.

To evaluate these cross ratios, see (2), Point 13.3.2. The difference is that the value then

found was –1 (a harmonic set), whereas now it is not; cross ratios on a and a ́ have to be just of

any equal value.

It is also pertinent to recall that 5 points define a conic: the center of a projective pencil

of lines and the four extremities of those lines.

Fig. 2 is an application of that explained hereinbefore, to a rectangle circumscribing an

ellipse. The centers of line pencils will be now the ellipse ́s vertices B and B ́ (given). Line a

will be the half-axis OA and line a ́, the segment AD.

Divide OA and AD in an equal number of parts (seven in the drawing) and fold both

line pencils. The intersection of corresponding fold lines (rounded in the drawing) are points of

the ellipse.

It ́s easy to check that both cross ratios are equal:

3

4

2

3

:

1

2

7

4

7

2

7

4

:^7

7

3

7

2

7

3

(^7) =
−
−
−
−

−
−
−
−
≡
a a
a a
a a
a a
OA
3
4
7
4
7
2
7
4
:^7
7
3
7
2
7
3
(^7) =
−
−
−
−
≡
b b
b b
b b
b b
AD
The verification has been made with the first four lines, but we might as well make the
set to carry on progressively.
We can get points in the other quadrants following the same procedure, or else, by
means of symmetry. Of course, the four vertices of the ellipse belong to it.
F ́ O
(^12)
V
V ́
a
a ́
B
B ́
A
D