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May 17th, 2009, 09:52 AM  #1 
Newbie Joined: Mar 2009 Posts: 12 Thanks: 0  Geometrical definition of a line: ax+by+c=0 vs. ax+by=1
A line is usually defined as: But why is it done like that, using 3 variables (a, b and c) instead of the simpler: It's more concise, it uses fewer variables (a and b). For example intersection with X and Y axis: Pretty much all such examples show the latter formula to give more concise results, basically because This are simple examples because we deal with 2dimensional lines, but it becomes more important as you go into bigger dimensions like planes in 3D being defined as instead of . Why is the more variables formula used by default instead of the simpler one? 
May 17th, 2009, 11:21 AM  #2 
Senior Member Joined: Jul 2008 Posts: 895 Thanks: 0  Re: Geometrical definition of a line: ax+by+c=0 vs. ax+by=1
There are several forms for most functions of this sort. Each serves a different purpose. Like a toolbox with an assortment of hammers, you can then choose the most appropriate for any particular problem. Your form ax + by = 1 is more commonly seen as the "Intercept form" x/a + y/b = 1. That is analogous to that form as in conic sections: x^2/a^2 + y^2/b^2=1, and x^2/a^2  y^2/b^2=1. These also have other forms, each suitable to a different purpose. When you get into vector algebra and lines, planes, and surfaces in 3D, you'll see different forms for the lines in space, again each serving a different purpose. "There's nothing new under the sun." 
May 17th, 2009, 11:41 AM  #3  
Newbie Joined: Mar 2009 Posts: 12 Thanks: 0  Re: Geometrical definition of a line: ax+by+c=0 vs. ax+by=1 Quote:
With ax+by=1 you can simply have b=1, a=0. Seems more correct, so why is that form more commonly used instead of this one for example. And where is the ax+by+c=0 form used? I looked on wikipedia but didn't find it used anywhere. What does the a, b and c represent, the slope, axis intersection position, or what? Thanks. BTW: I am currently trying to understand the subject better from a mathematical perspective, but I'll eventually use what I learn in programming, thus my interest for simplicity in functions. Fewer variables => smaller, faster code.  
May 17th, 2009, 04:24 PM  #4 
Global Moderator Joined: May 2007 Posts: 6,727 Thanks: 687  Re: Geometrical definition of a line: ax+by+c=0 vs. ax+by=1
ax+by+c=0 is the most general form. ax+by=1 is restrictive in that you can't have a line through the origin (y=ax).

May 17th, 2009, 05:51 PM  #5  
Senior Member Joined: Jul 2008 Posts: 895 Thanks: 0  Re: Geometrical definition of a line: ax+by+c=0 vs. ax+by=1 Quote:
In any event, all of this is not to be rehashed here [and that is all it amounts to ...rehashing] after centuries of development have already established useful forms. The only point is that anything that contains only a first degree relation will be linear: F = (9/5)C + 32.  

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