For the polar planimeter the " elbow " is connected to an arm with its other endpoint O at a fixed position.
12.
For the linear planimeter the movement of the " elbow " E is restricted to the " y "-axis.
13.
That means the planimeter measures the distance that its measuring wheel travels, projected perpendicularly to the measuring wheel's axis of rotation.
14.
That November, while attending a conference on computer graphics in Reno, Nevada, Engelbart began to ponder how to adapt the underlying principles of the planimeter to X-Y coordinate input.
15.
A direct development of the planimeter known as an integraph, or integerometer, can be used to establish the position of the centroid or center of mass of an irregular two-dimensional shape.
16.
I attacked her with proportional dividers, planimeter, rotameter, Simpson's rule, Froude's coefficients, Dixon Kemp's formulae, series, curves, differentials, and all the appliances of modern yacht designing, and she emerged from the ordeal a theoretically perfect boat.
17.
A mechanical device that computes area integrals is the planimeter, which measures the area of plane figures by tracing them out : this replicates integration in polar coordinates by adding a joint so that the 2-element linkage effects Green's theorem, converting the quadratic polar integral to a linear integral.
18.
Green's Theorem, which gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C, is applied in an instrument known as a planimeter, which is used to calculate the area of a flat surface on a drawing.
19.
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 ).
20.
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.
