Planimeter
A planimeter, also known as a platometer, is a measuring instrument used to determine the area of an arbitrary two-dimensional shape.
The Swiss mathematician Jakob Amsler-Laffon built the first modern planimeter in 1854, the concept having been pioneered by Johann Martin Hermann in 1818.
[1] Many developments followed Amsler's famous planimeter, including electronic versions.
At the end of one link is a pointer, used to trace around the boundary of the shape to be measured.
Near the junction of the two links is a measuring wheel of calibrated diameter, with a scale to show fine rotation, and worm gearing for an auxiliary turns counter scale.
As the area outline is traced, this wheel rolls on the surface of the drawing.
The operator sets the wheel, turns the counter to zero, and then traces the pointer around the perimeter of the shape.
When the tracing is complete, the scales at the measuring wheel show the shape's area.
When the planimeter's measuring wheel moves perpendicular to its axis, it rolls, and this movement is recorded.
The area of the shape is proportional to the number of turns through which the measuring wheel rotates.
The polar planimeter is restricted by design to measuring areas within limits determined by its size and geometry.
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.
The working of the linear planimeter may be explained by measuring the area of a rectangle ABCD (see image).
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 justification for the above derivation lies in noting that the linear planimeter only records movement perpendicular to its measuring arm, or when 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
For a parametric equation in polar coordinates, where both r and θ vary as a function of time, this becomes
