
Re: Which Polynomial?
Posted:
Aug 4, 2005 8:12 AM


A linear relation represents a "constraint" between two variables x and y (two degrees of freedom) is ax + by + c = 0 (a,b,c are constants). There is no need to keep three parameters, two are enough, for instance, dividing by b (b0), ux + y + v = 0 (u = a/b, v = c/b)
x,y,z were reserved for "point coordinates" while u,v,w were used for "linecoordinates". For instance, a straight line in the x,y plane is given by the equation y + ux + v = 0. This equation gives the united position of the line (u,v) and the point (x,y), i.e., the point lies on the line and the line goes through the point. So, (u) is the trigonometric tangent of the angle that the line makes with the xaxis and (v) is the yintercept. In the dual representation (interchanging line and point coordinates), the family of straight lines through the point (x,y) is given by v + xu + y = 0. So, given x and y, the equation represents the linear relation between "slope" and "yintercept".

