I'm working on some problem sets and I come across the phrases "P-symmetric" and "O-symmetric" which were referring to a region in the Cartesian plane. The only clue I have towards their meanings is one of the questions was asking if a region was P-symmetric "for some P in the plane". I would guess that O-symmetric is the same as P-symmetric but the point P must be the origin.
I searched for the two terms on the internet but I couldn't find anything. Are these phrases widely used in mathematics or could they have been devised specifically for these problem sets?
I'm not sure what these mean because I don't understand how a region could be symmetric about a point because if the region isn't a circle centered at the point, then not all parts of the region's boundary are equidistant from the point.
Just in case anyone is wondering, these problem sets are homework but finding the definitions of those two phrases is merely one part of a multi-step problem so I don't think this falls in the realm of asking for homework answers. I did ask my instructors about this but I know that I would probably not get an answer until tomorrow so that is why I am asking on here as well.
Best Answer
A region is symmetric about a point $P$ if it is preserved by a point reflection through $P$.
In 2-dimensional space, a point reflection is actually a $180^\circ$ rotation through $P$. So a region that has $180^\circ$ rotational symmetry is symmetric about some point in the plane, which I assume is what the term "$P$-symmetric" that you're seeing refers to.
(In higher dimensions, point reflections are not the same as rotations; the general definition is that a point reflection through $P$ maps a point $X$ to a point $X'$ such that $P,X,X'$ are collinear and $P$ is the midpoint of the segment $XX'$.)