Jesús de la Peña Hernández
x+y= 2 pλ and x−y= 2 pμ (2)
When assigning values to parameters λ or μ we get different planes that are vertical
(without z) and parallel to each other within either pencil of planes λ or μ because variables x,y
have equal coefficients, again, within both pencils.
The intersection of the hyperbolic paraboloid (1) with the planes (2) are straight lines
whose equations are:x+y= 2 pλ zp
x y
μ+ =λ system μ system (3)zp
x y
λ− = x−y= 2 pμIn turn, the first equation of λ system in (3) represents a secant vertical plane like A 1 AP 1
that is a vertical one. The second equation represents a plane such as VP 1 P 2 through the origin
(it lacks of the independent term).
Hence, plane A 1 AP 1 intersects the hyperbolic paraboloid along AP 1 that is one of the
straight lines composing it. That ́s why we can say that the hyperbolic paraboloid is a ruled sur-
face as seen in Fig. 2.
Till now, all the cuts have been done according to the vertical λ system. Something
alike happens with the parallel planes of the μ system (see Fig.2).
Summarising:- Through every point P 3 of the hyperbolic paraboloid (Fig. 2) pass two straight gen-
eratrices belonging to the λ and μ systems respectively. - Two generatrices of the same system cross to each other.
- Two generatrices of different systems meet at a point on the paraboloid.
- All the generatrices of the λ system (such as AP 1 ) are parallel to BCD. Something
analogous happens with system μ.
21.4.1 LAMINAR VERSION
Fig. 3 shows all the elements needed to construct a hyperbolic paraboloid by means of
their interlocking. The required slits will be performed from top to mid-point in Fig. 3.1, and
from the bottom to the mid-point in the trapeziums. Two corresponding cuts are shown as a
reference (see both pairs of scissors).
Note that each trapezium is duplicated: one is for the λ system, and the other (turned
over), is for the μ one. We can count 16 trapeziums besides the double rectangle of Fig. 3.2.
The trapeziums have vertical lap joints to interlock them and, in some cases, also hori-
zontal ones just to produce, whenever possible, a certain stiffness at the base of the secant
planes.
The secant planes to the hyperbolic paraboloid of Fig. 2 are: BP 1 V on parabola p; the
two perpendicular planes produced by Fig. 3.2 whose intersects are VP 4 (parallel to CD) and
VP5 (parallel to EC); the two viewed trapeziums BDCF and ECF; the rest of 16 trapeziums.
P 4 VP 5 is a horizontal right angle coincident with the half-asymptotes of a hyperbola
produced in the hyperbolic paraboloid when cut by a horizontal plane through V.
Fig. 4 shows the general appearance of the paraboloid though it lacks of some details.
The construction may not turn out to be perfect because of the interference between slits and
paper thickness. This handicap can be obviated when fine cardboard is used in connection with
wider slits. Anyhow, the figure, which is a beautiful one, resembles once more a wasp hive.