"Degrees of freedom" for lines
I'm reading something about "degrees of freedom" trying to learn what exactly it means, and there's this one sentence I'm running into that I can't really understand...
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They talk a lot about how a line on a plane is represented by the equation [tex]ax+by+c=0[/tex]. But I know from learning about [tex]y=mx+b[/tex] in grade school that you only need two numbers to specify a line.. :confused: If anybody could explain that sentence to me I'd really appreciate it. 
Re: "Degrees of freedom" for lines
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Re: "Degrees of freedom" for lines
{a:b:c} is shorthand for the proportion a/b= b/c. There are "two degrees of freedom" because you are "free" to choose two of the numbers to be almost anything you like and then could solve for the third.

Re: "Degrees of freedom" for lines
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Re: "Degrees of freedom" for lines
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Re: "Degrees of freedom" for lines
Well, you are not a caterpillar, are you? If you were constrained to a specific straight line, but could pick any point on that line, yes, that would be "one degree of freedom". Here, however, If we write a line as "ax+ by+ c= 0", we could multiply or divide each of the coefficients by any number (except 0 of course) and still have the same line: "rax+ rby+ rc= 0" is satisfied by exactly the same (x,y) and so is the same line. Notice that ra/rb= a/b and rb/rc= b/c no matter what r is. In the formula "ax+ by+ c= 0" two of the numbers can be chosen any way we want but the other is then fixed.

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