MATHEMATICS AND ORIGAMI

Mathematics and Origami

13 CONICS

As it ́s well known, they are curves determinated by the intersection of a cone and a plane not
passing through its vertex. Therefore they are plane curves as shown in Figs. 1, 2 and 3.

• Ellipse (Fig .1): The intersecting plane is oblique to the cone ́s axis in such a way that a close curve
  is obtained.


• Parabola (Fig. 2): It comes out as an open curve when the intersecting plane is parallel to one of the
  cone ́s generatrices. In turn, that generatrix is parallel to the parabola ́s axis.


• Hyperbola (Fig. 3): It is the other open conic, with two branches. The intersection plane has to be
  parallel to the cone ́s axis and cuts both cone ́s volumes: The two conventional cones have a com-
  mon vertex and their respective generatrices lie in opposition on the same straight line; that is what
  is called a complete cone. On intercepting plane of Fig. 3 we can see the asymptotes of the hyper-
  bola.

REMARKS ON CONICS AND ITS DEGENERATION

If we compare how ellipse and parabola are generated (Figs. 1 and 2), we can imagine that the
second is an ellipse whose unseen vertex is the ideal point of its axis
In case we revolve the plane of Fig. 1 till a position perpendicular to the axis of the cone, its
section in it becomes a circumference: this is, therefore, an ellipse with equal axes equivalent to the
diameter of said circumference.
If the plane passes through the vertex, the ellipse degenerates to one point, the parabola to a
generatrix and the hyperbola to a couple of coplanar generatrices.
Let ́s see here after the relations that can be issued between these curves and origami.

13.1 CIRCUMFERENCE
13.1.1 ITS CENTER

To obtain the center of a cut circumference given as in Fig.1, produce two cross-folds as

perpendicular as possible to each other, from edge to edge. The intersection of both folds will

be the center.

To produce it, look in Fig. 2 how point A becomes B because of the first symmetry, and

B becomes C after the second folding. Those symmetries make that OA = OB = OC. Thus we

have three points in a circumference equidistant from another interior one: this point is the

center according to circumference definition.

If the circle is not cut, but merely drawn on a piece of paper, the process is the same.

The only precaution is to have the circumference heavily marked to ease folding by transpar-

ency.