As stated above. I know that the $a, b, c$ represent the normal vector of the plane and that you can normalize them so that $a^2 + b^2 + c^2 = 1$. But what is the main reason for doing the normalization?
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Um, in which context? You'd have to ask the person who chooses to work with normalized equation in each case they make that choice, I think. – hmakholm left over Monica Sep 11 '18 at 16:30
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I am just reading a research paper on surface simplification (https://www.cs.cmu.edu/~./garland/Papers/quadrics.pdf), and they made that choice for selecting their planes. So I was just wondering what the point of doing normalization is... – Taylor Sep 11 '18 at 16:41
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In the paper, this is done in the context of minimizing the sum of squares of distances to this plane. The distance of a point $(x,y,z)$ from the plane $ax+by+cz+d=0$ is given by the formula $${|ax+by+cz+d|\over\sqrt{a^2+b^2+c^2}}.$$ If you use a unit normal for this plane, for which $a^2+b^2+c^2=1$, then this expression reduces to $|ax+by+cz+d|$, which then allows you to express the square of this distance as the square of the dot product of $[a,b,c,d]$ and $[x,y,z,1]$ as in equation (2) in the paper.
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