A sailing polar plot is a reference graph in polar coordinates used to determine the fastest speed a sailboat can travel given the wind speed and heading of the boat, relative to the apparent wind.
Polar coordinates are just another way to represent points on a graph. Instead of using (x,y) coordinates, the polar coordinate system uses an angle and distance away from the origin. Since the boat will be using information in the form of angles and magnitude, the polar coordinate system might be able to simplify calculations.
The graph on the left shows the wind coming from the top of the graph, each of the lines spiraling out from the boats position represent the optimal speed available at the corresponding angle. The further away the line is from the center, the greater the possible speed.
A boat's polars are typically created one of two ways: data collection or computational. Polars are unique to each kind of boat. Altering the rig, hull and keel design will vastly change the polars of the entire boat. It is important to note that a polars are a best case scenario; they assume flat water, constant wind speed and good sail trim.
Polars are most useful as a means of quantifying a boat's best possible velocity made good (VMG). VMG is the distance traveled in the direction of a target. For sailboats, it is very rare that a direct heading (ie sailing directly towards your intended target) has the best VMG. In effect, it is usually more efficient to steer a precise zig-zag path. Creating a polar plot for our boat would allow us to determine which direction the boat should point in order to achieve maximum VMG. Maximizing VMG results in the shortest time to get to a target. To create the polar plot, the boat needs to be sailed in varying wind conditions at different headings. The polar plot can be created by relating the apparent wind angle (AWA), apparent wind speed (AWS) and speed over ground (SOG).
If the polar data is stored in a lookup table or formula onboard, the boat could calculate it's most efficient heading. This defines the boat's tacking angles upwind, gybing angles downwind, and is useful for long distance navigation routing (a combination of both).