In a review technical report[1], it said the Hirt and Nicholos's alogrithm[2] for interface reconstruction is "piecewise constant/stair-step":

Page 6: "Piecewise constant/stair-step VOF methods followed shortly after SLIC, the most notable being Hirt and Nichols (HN) [2] (who coined the term 'volume-of-fluid')...In these methods, interfaces are also forced to align with mesh coordinates, but are additionally allowed to 'stair-step' within each cell, or align with more than one mesh coordinate within a given cell, depending upon the local volume fraction data distribution."

But when I look up [2], I found it's actually a PLIC (Piecewise Linear Interface Construction) algorithm:

Subsection D.3 Determing Interfaces Within a Cell: "...In the VOF technique, it's assumed that the boundary can be approximated by a straight line currint through the cell. By first determining the slope of this line, it can be moved across the cell to a position that intersects the known amount of fluid volume in the cell."

What's wrong?

Bibliography:

[1] Kothe, D.B; Rider, W.J., Comments on Modeling Interfacial Flows with Volume-of-Fluid Methods, Los Alamos National Lab, Feb 1995.

[2] Hirt,C.W; Nichols,B.D, Volume of Fluid (VOF) Method for the Dynamics of Free Boundaries, J.Comp.Phys., Vol.39, pp.201--225, 1981.

I have limited knowledge of SLIC techniques but I believe the slope was found but only used to distiguish the case between horizontal and vertical cases.

What's wrong? Don't believe everything you read, especially when one author is describing what another author did. Go to the original sources and find out for yourself.

(1). Why not try to locate both authors first, and then try to get their opinion. (2). We can't tell you what is right or what is wrong.