I have two parallel hyper planes $$a^Tx=b_1,a^Tx=b_2$$ where $a \in \mathbb{R}^n, x \in \mathbb{R}^n ,b \in \mathbb{R}$ and I want to find the distance between the two. I have read that the distance between the two hyperplanes is also the distance between the two points $x_1$ and $x_2$ where the hyperplane intersects the line through the origin and parallel to the normal vector $\vec a$. These points are given by $$x_1=\frac{b_1}{\|a\|^2_2}a$$ and $$x_2=\frac{b_2}{\|a\|^2_2}a$$ Then the distance is $|x_1-x_2|$ but I don't really understand how we got $x_1$ and $x_2$.
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1$\begingroup$ what is $\alpha$? $\endgroup$– uniquesolutionOct 17, 2015 at 10:42
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$\begingroup$ So what you have is an equation of a hyperplane in $\mathbb{R}^n$, that is, a translation of an $n-1$-dimensional subspace. There are lots of solutions. $\endgroup$– uniquesolutionOct 17, 2015 at 10:47
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$\begingroup$ I have not idea what you trying to do. You keep adding new information every time. You should take more care in formulating your problems. $\endgroup$– uniquesolutionOct 17, 2015 at 11:07
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$\begingroup$ @Marc van Leeuwen. Is there any difference between distance and the shortest distance between the two hyperplanes? $\endgroup$– holalaNov 28, 2020 at 3:41
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3 Answers
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Let $x_1$ be any point in the first hyperplane and consider the line $L$ that passes through $x_1$ in the direction of the normal vector $a$. An equation for $L$ is given by $x_1 + at$ for all $t\in\mathbb{R}$. Now find the intersection of $L$ and the second hyperplane:
$$ a^T(x_1 + a t) = b_2 \iff t = (b_2 - a^T x_1)/a^Ta = (b_2 - b_1)/a^T a $$
Therefore the intersection point is $x_2 = x_1 + a(b_2 - b_1)/a^Ta$. The distance between these two points is the distance between the hyperplanes:
$$ \|x_1 - x_2\| = \frac{|b_2 - b_1|}{a^Ta}\|a\| = \frac{|b_2-b_1|}{\|a\|} $$
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$\begingroup$ Just a small technical note, I think you meant to write $\vert \vert a \vert \vert^2$, since you need to take the square root to obtain the norm out of an inner product.... $\endgroup$– LouisMay 10, 2019 at 20:42
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1$\begingroup$ Hi Louis, in this case $a^Ta = \|a\|^2$. $\endgroup$ May 13, 2019 at 19:21
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$\begingroup$ @K.Miller Is there any difference between distance and the shortest distance between the two hyperplanes? $\endgroup$– holalaNov 28, 2020 at 3:42
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$\begingroup$ @holala Usually when one speaks about the distance between two geometric objects it is the shortest distance. $\endgroup$ Nov 28, 2020 at 16:44
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To try an other angle : $x_1 \in \mathbb R^n, a \in \mathbb R^n $
$$\left\| x_1 - a\right\| = \frac {b_1}{\left\| a \right\|} $$
If we consider a as the origin of $\mathbb R^n$ seen as an orthogonal plan we should be able to write $$ x_1 = \frac {b_1}{\left\| a \right\|} . \frac {a}{\left\| a \right\|} = \frac {b_1}{\left\| a \right\|^2}.a $$ and then your $x_1$.
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$\begingroup$ @Mc Cheng. Is there any difference between distance and the shortest distance between the two hyperplanes? $\endgroup$– holalaNov 28, 2020 at 3:43
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Points $x_1$ and $x_2$ are the images of the coordinate system's origin in an orthogonal projection on the two given planes.
