MATHEMATICS AND ORIGAMI

Jesús de la Peña Hernández

13.5 HYPERBOLA

This conic is the locus of the points such that the difference of the distances from them

to another two fixed points (focuses F, F ́), is constant. See Point 8.2.8.6.

Fig. 1 shows how to get a point P of a hyperbola and its tangent TT ́on it, from these

data: both focuses F, F ́, its center O and its director circle cd with center F and radius 2a.

This is the folding process:

• Bring F ́ to lie on any point A of cd: folding line, TT ́.


• To produce fold AF till the intersection with TT ́.


• Result: point P on the hyperbola and its tangent TT ́ on it.
  Proof:
  P is in the hyperbola for PF ́- PF = PA – PF = FA = 2a.
  Moreover, TT ́is the tangent on P because any other point but P (e.g. point G) does not
  fulfil the hyperbola ́s condition: GF ́ - GF = GA – GF > FA = 2 a (see ∆GAF).
  Fig. 2 shows how points and tangents of the lower part of right hand side branch of the
  hyperbola have been obtained. P ́ is the furthermost point. P ́ ́ is also in there and belongs to the
  left branch, which corresponds to focus F ́. Note that the points of the left branch are produced
  after the points on cd lying on its arc seen from F ́ (tangent F ́ T ́ ́ is one of its limits).
  Thus, there are two ways of constructing the left branch: take as many points in that
  seen arc as were taken in the rest of cd for the right branch, or else, do the same construction
  for circle cd ́with center F ́.
  Fig. 3 shows the complete hyperbola (rounded points) with its two branches.

F ́

cd

1

F

O

A

T ́

P

T

O

F ́

2

cd

F

P

A

T ́

T ́ ́ T

P ́ ́

P ́

G