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.
Read more about this topic: Planimeter
Famous quotes containing the word mathematical:
“It is by a mathematical point only that we are wise, as the sailor or the fugitive slave keeps the polestar in his eye; but that is sufficient guidance for all our life. We may not arrive at our port within a calculable period, but we would preserve the true course.”
—Henry David Thoreau (18171862)