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An example of a Poincaré section is all the points through which an orbiting body passes through some given plane, specifically passing through the plane in one general direction but not the other (e.g., the points where a planet passes through the ecliptic from south to north, but not those in which it passes through it from north to south). This is of interest if the orbit is shifting so that it does not repeatedly pass through the plane at the same point, and Poincaré sections provide a means of describing and analyzing long-term trends of such shifting orbits. The associated Poincaré map is a function mapping a chosen point of the Poincaré section to the next point, and each one to the next (but see below).
While the above example gives a two-dimensional "summary" of a three-dimensional process, in general, a Poincaré section can be created for any number of dimensions, the summary produced having a dimensionality smaller by one. The type of application-data is that in which a point (in whatever dimensional space) "moves" as time passes, i.e., the coordinates change over time.
The above example of an orbit's Poincaré map can be inconsistent: the orbit could intersect a plane at some specific point, and later intersect the same point on that plane, yet moving in some different specific direction, with the two subsequent circuits of the orbit crossing the plane at different subsequent points. By definition of a function, none would be able to take the role of its associated Poincaré map. To get around this, a Poincaré section to analyze an orbit can be formed using a 6-dimensional space that also includes three dimensions of velocity (or momentum), summarizing it in 5 dimensions.