Tic-Tac-Toe (井字遊戲)

Let f(a,b,x,y) = 1/exp(1/(a² - x²)²) + 1/exp(1/(a² - y²)²) - b

Equations:

Blue grid (#) ..... f(2, 0.999, x, y) = 0
Green nought (o) ..... f(0.125, 0.85, x, y) = 0

Red cross (x) ..... f(0.125, 0.999, 2(y+x), 2(y-x)) = 0

(Tessellations)

When the parameter 'a' changes from 0 to 2, the graph

changes shape from a plus sign(+) to a number sign(#).

In order to avoid discontinuities, we redefine f as follows:

f(a,b,x,y) = 1/exp(1/(s + (a² - x²)²)) + 1/exp(1/(s + (a² - y²)²)) - b,

where s(=0.001) is a small positive number.

Graph example:

• Click here to download the example file (tic_tac_toe.zip).

( Mathematical software used: Graph )