BACKGROUND: Spatial cognition research strives to maximize conditions favoring environment representation. This study examined how initial (egocentric) navigation headings interact with allocentric references in terms of world-based information (such as cardinal points) in forming environment representations. The role of individual visuo-spatial factors was also examined.
METHODS: Ninety-one undergraduates took an unfamiliar path in two learning conditions, 46 walked from cardinal south to north (SN learning), and 45 walked from cardinal north to south (NS learning). Path recall was tested with SN and NS pointing tasks. Perspective-taking ability and self-reported sense of direction were also assessed.
RESULTS: Linear models showed a better performance for SN learning and SN pointing than for NS learning and NS pointing. The learning condition x pointing interaction proved SN pointing more accurate than NS pointing after SN learning, while SN and NS pointing accuracy was similar after NS learning. Perspective-taking ability supported pointing accuracy.
CONCLUSIONS: These results indicate that initial heading aligned with cardinal north prompt a north-oriented representation. No clear orientation of the representation emerges when the initial heading is aligned with cardinal south. Environment representations are supported by individual perspective-taking ability. These findings offer new insight on the environmental and individual factors facilitating environment representations acquired from navigation.

Cardinal points are used for ray tracing through Gaussian systems. Anti-principal and anti-nodal points (which we shall refer to as the anti-cardinal points), along with the six familiar cardinal points, belong to a much larger set of special points. The purpose of this paper is to obtain a set of relationships and resulting equalities among the cardinal and anti-cardinal points and to illustrate them using Pascal\'s ring.
The methodology used relies on Gaussian optics and the transference T. We make use of two equations, obtained via the transference, which give the locations of the six cardinal and four anti-cardinal points with respect to the system. We obtain equalities among the cardinal and anti-cardinal points. We utilise Pascal\'s ring to illustrate which points depend on frequency and their displacement with change in frequency.
Pascal described a memory schema in the shape of a hexagon for remembering equalities among the points and illustrating shifts in these points when an aspect of the system changes. We modify and extend Pascal\'s ring to include the anti-cardinal points. We make use of Pascal\'s ring extended to illustrate which points are dependent on the frequency of light and the direction of shift of the equalities with change in frequency. For the reduced eye the principal and nodal points are independent of frequency, but the focal points and the anti-cardinal points depend on frequency. For Le Grand\'s four-surface model eye all six cardinal and four anti-cardinal points depend on frequency. This has implications for definitions, particularly of chromatic aberrations of the eye, that make use of cardinal points and that themselves depend on frequency.
Pascal\'s ring and Pascal\'s ring extended are novel memory schema for remembering the equalities among the cardinal and anti-cardinal points. The rings are useful for illustrating changes among the equalities and direction of shift of points when an aspect of a system changes. Care should be taken when defining concepts that rely on cardinal points that depend on frequency.