What are the two independent ratios used to specify a line's parameters?

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Discussion Overview

The discussion revolves around the concept of "degrees of freedom" in relation to specifying a line's parameters using the ratios {a : b : c}. Participants explore the implications of these ratios in the context of line equations, particularly the standard form ax + by + c = 0, and how they relate to the degrees of freedom in defining a line on a plane.

Discussion Character

  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions the meaning of "the two independent ratios {a : b : c}" and how it relates to the standard line equation, noting a discrepancy with the simpler form y = mx + b.
  • Another participant explains that the notation {a : b : c} represents the proportion a/b = b/c, indicating that two parameters can be freely chosen while the third is determined.
  • A different participant expresses confusion over the concept of degrees of freedom for a straight line, comparing it to a caterpillar on a wire, and asks for further clarification on how introducing more constants affects this concept.
  • One participant elaborates that while a caterpillar constrained to a line has one degree of freedom, the line's representation allows for two degrees of freedom because any non-zero constant can be multiplied to the coefficients without changing the line itself.

Areas of Agreement / Disagreement

Participants express varying levels of understanding regarding the notation and the concept of degrees of freedom. There is no consensus on the clarity of the notation {a : b : c} or its implications for degrees of freedom, indicating ongoing confusion and differing interpretations.

Contextual Notes

Some participants highlight the need to include vertical lines in the discussion, which may affect how degrees of freedom are understood in relation to the line equation. The discussion also reflects varying familiarity with mathematical notation and concepts.

sh86
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"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...
A line is specified by two parameters (the two independent ratios [tex]\{a : b : c\}[/tex]) and so has two degrees of freedom.

What is this "the two independent ratios {a : b : c}" ?

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.
 
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sh86 said:
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...


What is this "the two independent ratios {a : b : c}" ?

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.
The point that the author was trying to make is that to include ALL lines, you need to allow vertical lines (x=k). The form being used in the text allows for this (b=0). The two degrees of freedom is a way of saying that multiplying a,b,c by a constant doesn't change the line.
 


{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.
 


HallsofIvy said:
{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.

Wow, that {a:b:c} notation is confusing; I've never seen that.
 


flatmaster said:
Wow, that {a:b:c} notation is confusing; I've never seen that.

The notation is not new to me, but the concept of two degrees of freedom for a straight line is new (to me). I always thought a caterpillar walking along a wire had only one degree of freedom, same as all straight lines regardless of where they are. How does introducing more constants into the equation change that? Halls, can you expand a bit on your explanation?
 


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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