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I was reading this paper related to permutohedral lattice here. It mentions that the projection of a Real number in the $R^{d+1}$ space onto the hyperplane $H_d$ where

$ H_d = \{{\vec{x} | \vec{x}.\vec{1} = 0}\} \subset R^{d+1} $

T the projection of $R^{d+1}$ onto $H_d$ is given by $ T(\vec{x})= \vec{x}-\frac{(\vec{x}.\vec{1})}{(\vec{1}.\vec{1})}\vec{1} $

I am confused how the projection came to be like that. I think it should have been

$ T(\vec{x})= \frac{(\vec{x}.\vec{1})}{(\vec{1}.\vec{1})}\vec{1} $ isn't it?

They have mentioned that T is the projection of $R^{d+1}$ onto $H_d$ along $\vec{1}$. I didn't understand what along $\vec{1}$ meant. Projection is simply such that its projection on the plane is perpendicular to the normal of the plane isn't it. So I didn't get what this projection along this specific line means.

Any pointers

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As you have written the hyperplane $H_{d }$ consists of the set of $\vec{x}$ such that $\vec{x}\cdot \vec{1}=0$

Test this on the projection operator

$T(\vec{x})\cdot\vec{1}=(\vec{x}-\frac{\vec{x}\cdot \vec{1}}{\vec{1}\cdot \vec{1}}\vec{1})\cdot\vec{1}=\vec{x}\cdot\vec{1}-\frac{\vec{x}\cdot \vec{1}}{\vec{1}\cdot \vec{1}}\vec{1}\cdot\vec{1}=\vec{x}\cdot\vec{1}-\vec{x}\cdot\vec{1}=0$

as required.