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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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