MATHEMATICS AND ORIGAMI

Mathematics and Origami

13.4 PARABOLA

Although we already dealt with the parabola in Points 1.2.5 and 4 now we shall insist in

its properties and folding process.

As we know, the parabola is a conic whose points (P, P ́, etc.) are equidistant to a point

F (the focus, situated on the axis EF of the parabola) and to a line AD (directrix).

Here it is the folding process (Fig. 1) to get a parabola as the envelope of its tangents:

• To take a point (A, D, etc.) on the directrix.


• To produce lines AF. DF, etc.


• To fold the perpendicular bisectors PB, CH, etc.


• To produce the normals to the directrix through points A, D, etc. i.e. lines AP, DP ́,
  etc.


• To find the intersection points (P, P ́. etc.) of these normals and previous perpen-
  dicular bisectors PB, CH, etc.


• Lines PB, P ́C, etc. are the tangents to the parabola on points P, P ́, etc.
  It is easy to prove:
  Fold line CH (perpendicular bisector of FD) is an axis of symmetry, hence P ́ D = P ́F.
  Therefore P ́ is on the parabola.
  On the other hand, lying P ́ on CH, this fold CH has to be a tangent. This is so because
  any other of its points, e.g. P ́ ́ is at a smaller distance to the directrix than it is to the focus: P ́ ́I
  < P ́ ́D; in right triangle P ́ ́ I D, a leg is smaller than the hypotenuse.
  So, if P ́ is the only point on CH belonging to the parabola, it is because CH is the tan-
  gent to it on P ́.
  The process consists then in folding a given rectangle of paper in such a way that the
  points of its lower side (the directrix) are taken to lie on the focus F. The folds so produced are
  the tangents to the parabola (Fig. 2 also shows the rounded tangency points).
  Note that the vertex V, half way in between focus and directrix, is also a point of the pa-
  rabola, and the parallel VG to the directrix is the tangent to the conic on V.

(^12)
P
F P ́
A
B
V G
D
H
E
C
P E
P ́
F
A
B
G
C
P ́ ́
I
H