MATHEMATICS AND ORIGAMI

Mathematics and Origami

Now then, the 7 + 7 secants just obtained, are the intercepts in the main ellipse of plane
XY, produced by the circles (the cyclic sections) parallel both, to axis OZ and to t (or to its
symmetrical with respect to OX).

The plane tangent at U, as well as the three others, will produce cyclic sections of radius
zero.
If CC ́ is the trace of the cyclic section through the ellipsoid center, it means that CC ́ is
the diameter of a circumference whose points are all on the ellipsoid surface. Therefore OC is
the radius equivalent to the half-axis c we are after: c = OC.
To obtain the 14 cyclic sections it ́s enough to get the 4 corresponding to OV since the
rest are equal or symmetrical.
To get rid of confusion when interlocking the circles between each other, Fig. 3 shows
the traces of both pencils of planes: the circles should bear the corresponding number.

1

O A

B

x

y

t

U

1

1

V

V ́

2

O

1

B

y

1 x

A

C

C ́