MATHEMATICS AND ORIGAMI

Mathematics and Origami

21.4 HYPERBOLIC PARABOLOID

It is a quadric (a second order surface associated to one or various conics), that may be

generated this way (Fig.1):

• Begin with parabolas p and q that in turn may have equal or different parameters.


• They will have in common vertices and axes (OZ), while their foci are at either side
  of the common vertex.


• Their planes are at 90º.


• One of the parabolas, e.g. the p, will be the directrix; hence q will be the generatrix.


• Under these conditions, the hyperbolic paraboloid is generated when q moves par-
  allel to itself while resting all the time its vertex on p.


• Both parabolas p and q are the main sections of the paraboloid. Their common axis
  is also the axis of the paraboloid and the common vertex is its vertex too.

As can be seen, the hyperbolic paraboloid is an unlimited surface. In Figs. 1,2 and asso-

ciates is shown confined to a 90º sector (ECD of Fig. 2) of one of its two halves (the hyperbolic

paraboloid, like the parabola, has bilateral symmetry). To facilitate the representation we have

made p = q. Note that the altitude a in Fig. 1 is equal to FC in Fig. 2.

If we express as a vectorial relation the genesis of the hyperbolic paraboloid described

before (generatrix moving in parallel resting on the directrix), we get the equation of the

paraboloid:

z

q

y

p

x

2

2 2

− =

From now on we shall stick to the p = q simplification announced before. Hence, the

equation of our paraboloid becomes:

x^2 −y^2 = 2 pz (1)

This equation is referred to a set of co-ordinate axes like those in Fig. 1 but having its

origin at V.

To properly justify the construction of the hyperbolic paraboloid we are dealing with,

we shall cut it with several very special planes: they are vertical (i.e. parallel to axis Z) and par-

allel to each other:

Z

p

V P

X

Y

A

A

q

P

O

2

1

1

1

2

B

D

A

E

V

P

P

A

P

P

P

C

1

2 5

3

4

1

a

F