Instrument for determining the area (F) of an arbitrary two-dimensional shape (in red).
The blue bar can move only linearly in the direction perpendicular to the blue rollers axis.
Move stylus B along the periphery of the shape (one complete round), the green roller gives two values:
B1: initial position angle (in radians)
B2: final position angle (in radians)
F = L.R.(B1-B2)
L = AB
R: radius of rolling circle of the green roller.
The roller rotation axis must be parallel to AB.
Mathematical basis of the mechanism: Green’s theorem.
There must be sufficient friction between the green roller and the ground to prevent slipping.
Linear planimeters are used for the determination of stretched shapes.
In real planimeters there is reduction gear drive to ease reading angle values.