SLIC (Simple Line Interface Calculation) is an alternating-direction method for geometrically approximating fluid interfaces. It can be used in one, two, or three space dimensions and is characterized by the following features: (1) Fluid surfaces are represented locally in each mixed-fluid zone as a composition of one-dimensional components, one for each coordinate direction. (2) These one-dimensional components are composed entirely of straight lines, either perpendicular to or parallel to the coordinate direction. (3) The one-dimensional surface approximations for a mixed fluid cell are determined by testing whether the various fluids in the mixed cell are present or absent in the zone just to the left and right in the coordinate direction. (4) Because of the one-dimensional nature of the SLIC interface description, it is relatively easy to advance the fluid surfaces correctly in time. With the SLIC fluid-surface definitions, it should be possible to incorporate any one-dimensional method for advancing contact discontinuities, making SLIC very practical for numerical solutions of fluid dynamical problems. This paper addresses the problem of treating fluid interfaces in multifluid Eulerian hydrodynamic calculations. Two fundamental difficulties are identified: defining geometrical approximations to fluid interfaces and formulating Lagrangian equations of motion to advance these interfaces correctly in time. The discussion focuses on the former difficulty, although the simplicity of the SLIC surface approximations allows for a solution of the latter as well. Three methods are particularly noteworthy for calculating multifluid flow in a two-dimensional Eulerian context: the particle-in-cell (PIC) method, the coupled-Eulerian-Lagrangian (CEL) method, and the KRAKEN method. The SLIC method follows the KRAKEN approach, defining a local surface approximation for each mixed-fluid zone. However, it seeks the simplest possible interface description, leading to an alternating-direction method called SLIC. The SLIC rule for approximating surfaces prefers lines perpendicular to the coordinate axis over parallel lines. In three fluid zones where surfaces meet in a "Y-like" intersection, the intersection is approximated by a "T". Symmetry considerations ensure equal treatment of fluids when the same information is known about each. Surprisingly, limited information obtained by looking to the left and right in each coordinate direction, combined with the simplicity of straight-line surface approximations, is sufficient to construct a reasonable and workable representation of all possible fluid surface configurations in a mixed-fluid cell. The alternating-direction feature of SLIC makes it easily generalizable to more than three dimensions.SLIC (Simple Line Interface Calculation) is an alternating-direction method for geometrically approximating fluid interfaces. It can be used in one, two, or three space dimensions and is characterized by the following features: (1) Fluid surfaces are represented locally in each mixed-fluid zone as a composition of one-dimensional components, one for each coordinate direction. (2) These one-dimensional components are composed entirely of straight lines, either perpendicular to or parallel to the coordinate direction. (3) The one-dimensional surface approximations for a mixed fluid cell are determined by testing whether the various fluids in the mixed cell are present or absent in the zone just to the left and right in the coordinate direction. (4) Because of the one-dimensional nature of the SLIC interface description, it is relatively easy to advance the fluid surfaces correctly in time. With the SLIC fluid-surface definitions, it should be possible to incorporate any one-dimensional method for advancing contact discontinuities, making SLIC very practical for numerical solutions of fluid dynamical problems. This paper addresses the problem of treating fluid interfaces in multifluid Eulerian hydrodynamic calculations. Two fundamental difficulties are identified: defining geometrical approximations to fluid interfaces and formulating Lagrangian equations of motion to advance these interfaces correctly in time. The discussion focuses on the former difficulty, although the simplicity of the SLIC surface approximations allows for a solution of the latter as well. Three methods are particularly noteworthy for calculating multifluid flow in a two-dimensional Eulerian context: the particle-in-cell (PIC) method, the coupled-Eulerian-Lagrangian (CEL) method, and the KRAKEN method. The SLIC method follows the KRAKEN approach, defining a local surface approximation for each mixed-fluid zone. However, it seeks the simplest possible interface description, leading to an alternating-direction method called SLIC. The SLIC rule for approximating surfaces prefers lines perpendicular to the coordinate axis over parallel lines. In three fluid zones where surfaces meet in a "Y-like" intersection, the intersection is approximated by a "T". Symmetry considerations ensure equal treatment of fluids when the same information is known about each. Surprisingly, limited information obtained by looking to the left and right in each coordinate direction, combined with the simplicity of straight-line surface approximations, is sufficient to construct a reasonable and workable representation of all possible fluid surface configurations in a mixed-fluid cell. The alternating-direction feature of SLIC makes it easily generalizable to more than three dimensions.