# The Integraph: An Integrating Mechanism

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Before the Differential Analyzer and long before Mathematica* *or* *Wolfram|Alpha*,* mechanical instruments were used to perform computations. The "Integraph" [1] in this Demonstration plots the integral curve starting from a given curve .

Contributed by: Erik Mahieu (December 2015)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

The integral curve is drawn by a pen that moves in a (tangential) direction proportional to the local value of the given curve. This is achieved by the following mechanism.

The integraph consists of a rectangular carriage that moves left to right on rollers along the axis.

Drag the input pen along the given curve, following a given path . The input pen moves a trolley up and down the carriage and rotates the main arm around a fixed point on the carriage. The direction of the main arm is a function of the input .

with being the length of the main arm.

The output pen is guided parallel to the main arm by means of a tangential wheel connected to the main arm by a parallelogram linkage. This way the tangent of the output curve will be proportional to the input . This is expressed by the differential equation

,

with solution . The parallelogram keeps the wheel parallel to the main arm, independent of the vertical position of the output pen.

References

[1] Wikipedia. "Integraph." (Nov 20, 2015) en.wikipedia.org/wiki/Integraph.

[2] Wikipedia. "Bruno Abakanowicz." (Nov 20, 2015) en.wikipedia.org/wiki/Bruno_Abakanowicz.

[3] G. Coradi, "Intégraphes Nos. 40 et 41," *Original-Katalog*, No. 37, p. 12. www.g.coradi.com.

[4] P. Garcia. "Integraphs," DrawingMachines.org. (Dec 4, 2015) drawingmachines.org/category.php?id=36.

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