MATHEMATICS AND ORIGAMI

Mathematics and Origami

21 QUADRICS

They are conic generated surfaces (see Point 13). That generation is associated always

to a combination of conics in the role of directrices and generatrices, respectively.

They have several variants; among them we find the round bodies (Point 19) that may

be considered as degenerated quadrics. The basic quadrics are the ellipsoid, the hyperboloid

and the paraboloid. From here on we shall study:

• Two types of ellipsoid, both interlocked laminar: the elliptic one, strictly speaking,
  and the other, also elliptic but deformable and made up of cyclic sections.


• The revolution ruled hyperboloid, considered as a virtual surface (see Point 17).


• The hyperbolic paraboloid, also in two versions: a virtual surface and as a deform-
  able interlocked laminar construction.

21.1 ELLIPTIC ELLIPSOID

Fig.1 shows the wire-work version of the ellipsoid where one can see the three ellipses

with vertices A, B; A, C; B, C that intersect orthogonally at the quadric center.

Note that the three axes have different length, hence the resultant ellipsoid is an elliptic

one. Had two of the axes been equal to each other, one out of the three ellipses would had be-

come a circumference and therefore the quadric would be an ellipsoid of revolution.

The ellipsoid generation takes place when the ellipses with vertices AC and BC act as

directrices, and a horizontal one, acting as generatrix, moves in a parallel direction resting on

the other two. The latter is shown in different positions: a centered one and two others symmet-

ric to it.

Fig. 2 is a laminar vision of the paper-constructed ellipsoid. It consists in a total of 9 el-

lipses: the three main ones, two parallel to plane AB and four parallel to plane BC. The three

horizontal ellipses have to be made in halves to permit assembly. Fig. 3 has all the ellipses or

half-ellipses needed to build up the ellipsoid.