Understanding terminology related to implementing the sobel edge detection algorithm I'm trying to follow a scholarly paper that discusses a modified implementation of the sobel edge detection algorithm, but I'm not following the terminology 100%.

The Sobel detector consists of four directional gradient masks: 
  horizontal ($S_H$), vertical ($S_V$), left diagonal ($S_{LD}$), and
  right diagonal ($S_{RD}$). We calculate the edge strength ($S$) using
  ($1$) to highlight both edges and corners as follows:

$$ S = MAX(|S_H|,|S_V|, |S_{LD}|, |S_{RD}|) + k * |S_{\perp - MAX}|\space\space\space\space\space\space(1)$$

where $_{\perp - MAX}$ is normal to the maximum gradient direction and $k$ is set to be 0.5

The typical implementation of the sobel algorithm uses only the vertical and horizontal gradients. To calculate a pixel, take the dot product of the pre-defined sobel operator (3x3 matrix) with the pixel and all it's neighboring pixels (3x3), sum up the values, then take the magnitude of the 2 (or in this case 4) gradients:
$$pixel(x,y) = \sqrt{{S_{H}}^2+{S_V}^2}$$
My confusion with equation ($1$) is what's meant by MAX, and "normal to the maximum gradient". 
By "MAX gradient" do they just refer to the magnitude of the 2 (or in this case 4) gradient values (the root of the sum of squares)? And by normal, I know how to take the normal of a 2-dimensional value (or 4-dimensional), such as is produced by $[ S_H \space\space S_V ]$, if that is the meaning.
I'm just not confident that I'm reading this correctly though.

First you compute the four values $|S_H|$, $|S_V|$, $|S_{LD}|$, $|S_{RD}|$. Then you pick the largest of these; this is the first summand. Then you pick the orthogonal direction, that is, if $|S_H|$ was the maximal value, take $|S_V|$; if $|S_{LD}|$ ws maximal, take $|S_{RD}|$; and so on. This is the $S_{\perp-MAX}$ used in the second summand. (This definition does have a problem though if for example $|S_H|=|S_{RD}|$ are both maximal and the corrsponding orthogonal gradients $|S_V|, |S_{LD}|$ differ).
