Formula of signed distance from hyperplane to point Let $H$ be a hyperplane defined by the points $p_1, p_2, ..., p_n$ and single point $x$ generally out of the hyperplane. Is there any formula to calculate the signed distance between $x$ and $H$?
I found, that following value is the orientation (i.e. its sign matches the sign of the required signed distance):
$\displaystyle
\Delta_{H}(x)=
\left\vert
\begin{array}{ccccc}
 p_{1,1} & p_{1,2} & \cdots & p_{1,n} & 1 \\
 p_{2,1} & p_{2,2} & \cdots & p_{2,n} & 1 \\
 \vdots & \vdots & \vdots & \vdots & \vdots \\
 p_{n,1} & p_{n,2} & \cdots & p_{n,n} & 1 \\
 x_{1} & x_{2} & \cdots & x_{n} & 1
\end{array}
\right\vert$
But I suspect, that $\Delta_H(x)$ have a bound with value of signed distance.
Am I right? What is the real formula of the signed distance from hyperplane to point?

Finally, I found the solution (in book Andrew J. Hanson: Geometry for N-Dimensional Graphics):
$
\displaystyle
V=\frac{1}{n!}
\left\vert
\begin{array}{ccccc}
 p_{1,1} & p_{1,2} & \cdots & p_{1,n} & 1 \\
 p_{2,1} & p_{2,2} & \cdots & p_{2,n} & 1 \\
 \vdots & \vdots & \ddots & \vdots & \vdots \\
 p_{n,1} & p_{n,2} & \cdots & p_{n,n} & 1 \\
 x_{1} & x_{2} & \cdots & x_{n} & 1 \notag
\end{array}
\right\vert
\\
S=\frac{1}{(n - 1)!}
\cdot
\sqrt{
\left\vert
\left ( 
\begin{array}{ccccc}
 p_{1,1} & p_{1,2} & \cdots & p_{1,n} & 1 \\
 p_{2,1} & p_{2,2} & \cdots & p_{2,n} & 1 \\
 \vdots & \vdots & \ddots & \vdots & \vdots \\
 p_{n,1} & p_{n,2} & \cdots & p_{n,n} & 1 \notag
\end{array}
\right )
\cdot
\left ( 
\begin{array}{ccccc}
 p_{n,1} & p_{2,1} & \cdots & p_{1,1} \\
 p_{n,2} & p_{2,2} & \cdots & p_{1,2} \\
 \vdots & \vdots & \ddots & \vdots \\
 p_{n,n} & p_{2,n} & \cdots & p_{1,n} \\
 1 & 1 & \cdots &1 \notag
\end{array}
\right )
\right\vert
}
\\
D=\frac{n\cdot V}{S}
$
