the movement of the "elbow" E is restricted to the y-axis. For the polar planimeter the "elbow" is connected to an arm with its other endpoint O at a fixed position. Connected to the arm ME is the measuring wheel with its axis of rotation parallel to ME. A movement of the arm ME can be decomposed into a movement perpendicular to ME, causing the wheel to rotate, and a movement parallel to ME, causing the wheel to skid, with no contribution to its reading. Principle The working of the linear planimeter may be explained by measuring the area of a rectangle ABCD (see image). Moving with the pointer from A to B the arm EM moves through the yellow parallelogram, with area equal to PQ×EM. This area is also equal to the area of the parallelogram A"ABB". The measuring wheel measures the distance PQ (perpendicular to EM). Moving from C to D the arm EM moves through the green parallelogram, with area equal to the area of the rectangle D"DCC". The measuring wheel now moves in the opposite direction, subtracting this reading from the former. The movements along BC and DA are the same but opposite, so they cancel each other with no net effect on the reading of the wheel. The net result is the measuring of the difference of the yellow and green areas, which is the area of ABCD. Mathematical derivation The operation of a linear planimeter can be justified by applying Green's theorem onto the components of the vector field N, given by: where b is the y-coordinate of the elbow E. This vector field is perpendicular to the measuring arm EM: and has a constant size, equal to the length m of the measuring arm: Then: because: The left hand side of the above equation, which is equal to the area A enclosed by the contour, is proportional to the distance measured by the measuring wheel, with proportionality factor m, the length of the measuring arm. The justification for the above derivation lies in noting that the linear planimeter only records movement perpendicular to its measuring arm, or when is non-zero. When this quantity is integrated over the closed curve C, Green's theorem and the area follow. Polar coordinates The connection with Green's theorem can be understood in terms of integration in polar coordinates: in polar coordinates, area is computed by the integral where