Polygon Equation

Research archive · original page Research.PolygonEquation · last recorded update 2006-10-28

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

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

1. Here is an infinite line: x+y-1=0.
2. Here is an alternate form: (x+y-1)'^2^'=0 Note that the left hand side cannot be negative.
3. Here is an equation of a curve that satisfies either of two conditions A(x,y)'^2^'=0 or B(x,y)'^2^'=0: A(x,y)'^2^' B(x,y)'^2^' = 0
4. Here is an equation of a curve that satisfies both conditions A(x,y)'^2^'=0 and B(x,y)'^2^'=0: A(x,y)'^2^' + B(x,y)'^2^' = 0
5. 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
6. 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
7. 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.