MATHEMATICS AND ORIGAMI

Mathematics and Origami

7.14 FUNDAMENT OF ORTHOGONAL BILLIARDS GAME (H. HUZITA)

This game is an ingenious recreation of conventional billiards. In this, any ball cast against the

tableside is rebounded out in such a way that angles of incidence and reflexion are congruent

(Fig. 1).

In orthogonal billiards, this other hypothesis is set up: once the ball hits the tableside, it is al-

ways repelled in a direction normal to incidence (Fig. 2).

As we shall see, HH ́s hypothesis is very useful to solve different problems (geometric as well

as algebraic). Let ́s see first, how balls behave under each of the following conditions.

Play to one tableside, only

In conventional billiards (Fig. 3), when we hit the white ball B against the tableside to reach

sen(α γ) senγ

ZV ZB

=

+

;

sen(π α (π γ)) sen(π−γ)

=

− − −

ZV ZR

Equalising ZV:

γ

γ α

γ

α γ

sen

sen( )

sen

sen( ) −

=

+

ZB ZR ; developing:

(ZR+ZB)senαγcosγ=(ZR−ZB)cosαsen

γ tgα

2

tg

BR

ZB+BR

=

Contrarily, in orthogonal billiards there may be one, two or none solutions (Fig. 4) depending

on the fact that the circumference with diameter BR will be tangent, secant or will not reach the

tableside.

(^12)
red ball R, there is always one solution:
Data are BZ; RZ; a; γ is the un-
known.
Being ZV common in ∆ZVB and
∆ZVR, we have:
3
Z
B
R
V