Mathematics and Origami

From that drawing we get the angle of the hyperboloid ́s asymptotic cone as

= 2 arctg = 84 , 7369 º

b

a

α

Note that the angle is smaller than that of the cones in Fig. 6 (100,57º). This has being
designed purposely to elude that the vertices of both cones will get in contact before their sur-
faces do on the hyperboloid bases.

To finish, let ́s set forth the question of what the tangent plane on a point of the hyper-

boloid will be.

For that we shall do some simple changes in equation (2):

2 2 2

(^1) 





 = −





 −





a
y
b
z
a
x
(difference of squares)





 −





= +





 −





 +
a
y
a
y
b
z
a
x
b
z
a
x
1 1
u
b
z
a
x
a
y
a
y
b
z
a
x

• 
  


• 
  

  =
  −
  − 1
  1
  ; v
  b
  z
  a
  x
  a
  y
  a
  y
  b
  z
  a
  x

  
  


• 
  

  −

  
  


• 
  

  − 1
  1
  (4)
  0
  1 1

  
  


• − z−u=
  b
  y
  a
  u
  x
  a
  0
  1 1
  − − z−v=
  b
  y
  a
  v
  x
  a
  (5) (6)
  1 0
  1
  − + z− =
  b
  u
  y
  a
  x
  a
  u
  1 0
  1 1


• 
  
  
  
  • z− =
    b
    y
    a
    x
    a
    v
    Finally we get equations (5) and (6) that mean:
  
  
  
  

• The four of them are equations of planes.


• Either pair (5) or (6) represent the intersection lines of each of those two planes.


• Equations (5) are parametric in u, and (6) in v, what means that for each value of u
  we get a straight line (5) and for values v the right line we obtain is (6).


• Do not mix up lines u, v with the plückerian co-ordinates of Point 13.2. If we de-
  velop equations (5) and (6) (what we shall not do), we would arrive to the asymp-
  totic or hiperboloidal co-ordinates that define parametrically the hyperboloids.


• Getting a new generatrix for each value of u recalls what was said earlier when each
  rotation of an angle of 17,142857º yielded, also, another generatrix.


• Generatrices v are the symmetric of u with respect to a plane containing the hyper-
  boloid axis and the mid-point of a generatrix u. This way, Fig. 11 has been drawn
  from Fig. 4.


• Note that to get Fig. 11 as a paper construction, it suffices to begin with a Fig. 1
  changed in such a way that the mountain / valley crease would be symmetric with