My Math Forum Geometrical definition of a line: ax+by+c=0 vs. ax+by=1

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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: $ax+by+c=0$ But why is it done like that, using 3 variables (a, b and c) instead of the simpler: $ax+by=1$ It's more concise, it uses fewer variables (a and b). For example intersection with X and Y axis: $OX: y=\frac{1}{b} and OY: x=\frac{1}{a}; \\ vs. \\ OX: y=-\frac{c}{b} and OY: y=-\frac{c}{a};$ Pretty much all such examples show the latter formula to give more concise results, basically because $ax+by+c=0 \Rightarrow -\frac{a}{c}x-\frac{b}{c}y=1\\ \mbox{ so we get the simpler formula } a'x+b'y=1 \mbox{ with } a'=-\frac{a}{c} \mbox{ and } b'=-\frac{b}{c}$ This are simple examples because we deal with 2-dimensional lines, but it becomes more important as you go into bigger dimensions like planes in 3D being defined as $ax+by+cz+d=0$ instead of $ax+by+cz=1$. 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
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Re: Geometrical definition of a line: ax+by+c=0 vs. ax+by=1

Quote:
 Originally Posted by Dave 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."
But why do they use division in x/a + y/b = 1, it doesn't seem to be a very good form. For example to create a horizontal line with y=1 you'd have to have b=1 and a=infinity.

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
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Re: Geometrical definition of a line: ax+by+c=0 vs. ax+by=1

Quote:
 Originally Posted by manixrock But why do they use division in x/a + y/b = 1, it doesn't seem to be a very good form.
It is precisely YOUR form, but is more descriptive. If you wrote yours as Ax + By = 1, your A = my 1/a, and your B = my 1/b.

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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