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I was reading through a geometry book for computer vision and it presented that the homogeneous representation of lines is

$$ax+by+c =0 \Leftrightarrow (a,b) \neq 0$$

But then they introduced an example that says

Consider the two lines $x=1$ and $x=2$. Here the two lines are parallel and consequently intersect "at infinity". In homogeneous notation the lines are $l=(-1,0,1),\ l'=(-1,0,2)$.

how do they get the values of $a$, $b$, and $c$ given the equations $x=1$ and $x=2$ ?

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  • $\begingroup$ @Hee Kwon Lee $(a,b,c)$ is not a zero vector. $\endgroup$ Dec 24, 2013 at 2:54

1 Answer 1

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$a$, $b$, and $c$ are the homogeneous coordinates of a line defined by $ax+by+c=0$.

$x=1$ implies $-x+1=0$ and $a=-1,b=0,c=1$. The same for the other line.

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  • $\begingroup$ Wouldn't $x=1$ also imply $x-1=0$ therefore, $a=1$, $b=0$ and $c=-1$. Is this actually correct? $\endgroup$
    – BRabbit27
    Dec 16, 2013 at 12:22
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    $\begingroup$ Yes, it's a property of the homogeneous coordinates. If you multiply all of them by the same non-zero number, you still get the same line, e.g. $kax+kby+kc=0$ is the same line as $ax+by+c=0$. $\endgroup$ Dec 16, 2013 at 20:46

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