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Topic: Which Polynomial?
Replies: 28   Last Post: Jul 27, 2006 3:13 PM

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Posts: 59
Registered: 1/30/05
Re: Which Polynomial?
Posted: Aug 4, 2005 8:12 AM
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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 (b‚0), 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
"line-coordinates". 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 x-axis and (-v) is the y-intercept. 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 "y-intercept".

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