MATHEMATICS AND ORIGAMI

Jesús de la Peña Hernández

13.3 ELLIPSE

13.3.1 TO FIND ITS PARAMETERS

Let loose ellipse e on paper p; it may be a CAD print. Folding process will be as fol-

lows.

1. Likewise circumference (Point 13.1.1):
   A → A ́ ; B → B ́
   Result: axes AA ́, BB ́ and consequently center O.


2. C → AA ́ ; A → A
   D → AA ́ ; A ́ → A ́
   E → BB ́ ; B → B
   G → BB ́ ; B ́ → B ́
   Result: circumscript rectangle to the ellipse.


3. H → AA ́ ; B ́→ B ́
   Result: focus F.


4. F → OA ́ ; O → O
   Result: second focus F ́.

Note that in the ∆B ́OF obtained in step 3, it is:

B ́F^2 =B ́H^2 =a^2 =OB ́^2 +OF^2 ; a^2 =b^2 +c^2

13.3.2 THE ELLIPSE AS THE ENVELOPE OF ITS OWN TANGENTS

The given data are:

• The measure of its half-axis a.


• The position of its focuses F ́, F on the plane.
  That amounts to a given circumference with radius 2a and center F ́ (director circle cd), and
  the position of F inside it: distance FF ́ is equivalent to 2c (Fig. 1).
  Likewise, the principal circle cp (Fig. 3) is given: it has O as center and a as radius.
  Here we have the folding process to get the tangents of Fig. 2:
  Take any point P on the circumference, to lie on F by fold AB: AB will be the tangent to the
  ellipse. Then fold F ́P: the intersection point T of both folds is the tangency point.
  This is the proof:
  F ́T + TF = F ́T +TP = 2a (ellipse condition)
  So T is a point on the ellipse. For any other point of AB, e.g. S we have:
  F ́S + SF = F ́S + SP > F ́P = 2a (∆F ́SP)

p

e

A

A ́

B

B ́

C

D

E

G

A

B ́ A ́

B

F

F ́

H

p

O