Hart's Inversor
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 OPBD. Similarly, in ΔABC, AQAC. Because of the symmetry, or since
The following property of the configuration will be proved later:
(2)  OP·OQ = m(1  m)·(AD^{2}  AB^{2}). 
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.)
(4) 

However, by the Pythagorean theorem,
ED^{2} + AE^{2} = AD^{2} and EB^{2} + AE^{2} = AB^{2}. 
Hence from (4),
(5)  AC·BD = AD^{2}  AB^{2}. 
Further
(6) 
OP/BD = AO/AB = m and OQ/AC = OB/AB = 1  m. 
Combining (5) and (6) we obtain

References
 R. Courant and H. Robbins, What is Mathematics?, Oxford University Press, 1996
 H. Rademacher and O. Toeplitz, The Enjoyment of Mathematics, Dover, 1990
Inversion  Introduction
 Angle Preservation Property
 Apollonian Circles Theorem
 Archimedes' Twin Circles and a Brother
 Bisectal Circle
 Chain of Inscribed Circles
 Circle Inscribed in a Circular Segment
 Circle Inversion: Reflection in a Circle
 Circle Inversion Tool
 Feuerbach's Theorem: a Proof
 Four Touching Circles
 Hart's Inversor
 Inversion in the Incircle
 Inversion with a Negative Power
 Miquel's Theorem for Circles
 Peaucellier Linkage
 Polar Circle
 Poles and Polars
 Ptolemy by Inversion
 Radical Axis of Circles Inscribed in a Circular Segment
 Steiner's porism
 Stereographic Projection and Inversion
 Tangent Circles and an Isosceles Triangle
 Tangent Circles and an Isosceles Triangle II
 Three Tangents, Three Secants
 Viviani by Inversion
 Simultaneous Diameters in Concurrent Circles
 An Euclidean Construction with Inversion
 Construction and Properties of Mixtilinear Incircles
 Two Quadruplets of Concyclic Points
 Seven and the Eighth Circle Theorem
 Invert Two Circles Into Equal Ones
Activities Contact Front page Contents Geometry
Copyright © 19962018 Alexander Bogomolny
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