Planimeter

3 years ago by in Featured, Geometry, Geometry

7planimeter (also known as a platometer) is a measuring instrument used to determine the area of an arbitrary two-dimensional shape.

Principle of the linear planimeter

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 net result is the measuring of the difference of the yellow and green areas, which is the area of ABCD. There are of course the movements along BC and DA, but as they are the same but opposite, they cancel each other on the reading of the wheel.

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:

\!\,N(x,y)=(b-y,x),

where b is the y-coordinate of the elbow E.

This vector field is perpendicular to the measuring arm EM:

\overrightarrow{EM}\cdot N=xN_x+(y-b)N_y=0

and has a constant size, equal to the length m of the measuring arm:

\!\,\|N\|=\sqrt{(b-y)^2+x^2}=m

Then:

\oint_C(N_xdx + N_ydy)=\iint_S\left(\frac{\partial N_y}{\partial x}-\frac{\partial N_x}{\partial y}\right)dxdy=
=\iint_S\left(\frac{\partial x}{\partial x}-\frac{\partial (b-y)}{\partial y}\right)dxdy=\iint_S dxdy=A,

because:

\frac{\partial}{\partial y}(y-b)=\frac{\partial}{\partial y}\sqrt{m^2-x^2}=0,

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.

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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 \scriptstyle \int_\theta \tfrac{1}{2} (r(\theta))^2\,d\theta, where the form being integrated is quadratic in r, meaning that the rate at which area changes with respect to change in angle varies quadratically with the radius.

For a parametric equation in polar coordinates, where both r and θ vary as a function of time, this becomes

\int_t \tfrac{1}{2} (r(t))^2 d(\theta(t))=\int_t \tfrac{1}{2} (r(t))^2\, \dot \theta(t)\,dt.

For a polar planimeter the total rotation of the wheel is proportional to \scriptstyle \int_t r(t)\, \dot \theta(t)\,dt, as the rotation is proportional to the distance traveled, which at any point in time is proportional to radius and to change in angle, as in the circumference of a circle (\scriptstyle \int r\,d\theta=2\pi r).

This last integrand \scriptstyle r(t) \,\dot \theta(t) can be recognized as the derivative of the earlier integrand \scriptstyle \tfrac{1}{2} (r(t))^2 \dot \theta(t) (with respect to r), and shows that a polar planimeter computes the area integral in terms of the derivative, which is reflected in Green’s theorem, which equates a line integral of a function on a (1-dimensional) contour to the (2-dimensional) integral of the derivative.

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