Does the (piecewise straight border) line of a polygon have an equation, as a circle has the equation *x ^{2}+y^{2}=1*?

Yes, but I don't know why you'd want to use it. We'll approach the answer in steps, by examples.

- Here is an infinite line:
*x+y-1=0*. - Here is an alternate form:
*(x+y-1)*Note that the left hand side cannot be negative.^{2}=0 - Here is an equation of a curve that satisfies either of two conditions
*A(x,y)*or^{2}=0*B(x,y)*:^{2}=0*A(x,y)*^{2}B(x,y)^{2}= 0 - Here is an equation of a curve that satisfies both conditions
*A(x,y)*and^{2}=0*B(x,y)*:^{2}=0*A(x,y)*^{2}+ B(x,y)^{2}= 0 - Here's how to turn the inequality
*x>=0*into an equation.*q*is a variable that is not used anywhere else.*q*^{2}= x - Here is a finite segment, or edge, of the above line, for
*0<=x<=10*:*(x+y-1)*^{2}+ (q^{2}-(25-(x-5)^{2})) = 0 - Now you have enough to find the equation for a whole polygon.

The reason that the above works is that these are complex curves that have small regions that are real.