Similar to Peaucellier linkage, Hart's linkage (Hart's inversor or Hart's cell) employs inversion to convert between circular and rectilinear motion. Compared to the Peaucellier linkage, Hart's device uses fewer rods.
The device consists of four rods AB, BC, CD, and AD, such that
|(1)||AO/AB = AP/AD = CQ/BC = m,|
for 0 < m < 1. In ΔABD, (1) implies OP||BD. Similarly, in ΔABC, AQ||AC. Because of the symmetry, or since
The following property of the configuration will be proved later:
|(2)||OP·OQ = m(1 - m)·(AD2 - AB2).|
It indicates that P and Q are mutually inverse under an inversion with center O. This means that, if O is fixed and P traces a curve, Q will trace the inverse image of the curve. If an additional rod SP is so attached to the configuration that
|(3)||SP = SO|
and S is fixed, then P will trace a circle that passes through the center O of inversion. It follows that Q will then describe a segment of a straight line.
The applet below demonstrates this property. The points A, B, D, O, and S are draggable for the purpose of defining (or redefining) the attributes of the configuration. However, when P is dragged both O and S remain fixed.
Note that the dimensions of the rods impose certain limitations on the relative positions of the rods. When these are about to be violated while P is being dragged, the applet stops tracing the points. If this happens, return P into the arc already drawn and reconfigure from here.
|What if applet does not run?|
Let's now prove the claim (2), as promised.
Draw AE and CF perpendicular to BD (hence also to AC.)
However, by the Pythagorean theorem,
ED2 + AE2 = AD2 and|
EB2 + AE2 = AB2.
Hence from (4),
|(5)||AC·BD = AD2 - AB2.|
OP/BD = AO/AB = m and|
OQ/AC = OB/AB = 1 - m.
Combining (5) and (6) we obtain
- R. Courant and H. Robbins, What is Mathematics?, Oxford University Press, 1996
- H. Rademacher and O. Toeplitz, The Enjoyment of Mathematics, Dover, 1990
Copyright © 1996-2018 Alexander Bogomolny