Suppose I have two parallel hyperplanes: $\{x \in \mathbb{R}^n|a^Tx=b_1\}$ and $\{x \in \mathbb{R}^n|a^Tx=b_2\}$. How would I find the distance between these two hyperplanes?
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The minimal distance would be between points with difference parallell to $a$ sou we would have that the points could be written $x$ and $x+ta$ and then solve for $t$:
$$b_1 + t|a|^T = a^Tx + a^Tta = a^T(x+ta) = b_2$$
So $t = (b_2-b_1)/|a|^2$, then insert that into the difference of the points $ta$ you'd get:
$$|ta| = |b_2-b_1|/|a|$$
skyking
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The linear form $a^T:\Bbb R^n\to\Bbb R$ takes constant values on any plane parallel to your hyperplanes, and $b_1,b_2$ give the respective values of the linear form on those planes themselves. The distance you are looking for is than clearly proportional to $|b_2-b_1|$, and the factor of proportionality can be found by taking a special case where the distance is known: take $b_1=0$, and $b_2=a^Ta=\|a\|^2$, the planes respectively going through the origin and the normal vector $a$. For this case $|b_2-b_1|=\|a\|^2$, while the distance is the length $\|a\|$ of $a$; the right formula for the distance then must be $$ d=\frac{|b_2-b_1|}{\|a\|}.$$
Marc van Leeuwen
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