Peaucellier-Lipkin Linkage

How the Peaucellier linkage works

Start with a single fixed pin in the table — call it O. From O run two rigid arms of equal length m, pinned at O but free to swing. Their far ends are two opposite corners of a rhombus — a four-bar loop whose four sides are all the same shorter length l. Label the other two corners of the rhombus P and Q. That is the whole "cell": two long arms, four short bars, one fixed pin. Six bars, and we have not drawn anything yet.

Here is the geometry that makes it magic. Because the figure is symmetric about the line through O, P, and Q, those three points are always collinear, no matter how you flex the mechanism. And the product of the distances OP × OQ is a constant — it never changes as the linkage moves. P and Q are locked into being inverses of each other about a circle centered at O. Move P closer to O and Q swings out; push P out and Q is reeled in, but the product of their distances stays pinned to one number.

So far the cell is just an inversor: feed it any motion of P and it spits out the inverse motion of Q. To draw a straight line we add one final bar — bar number seven, a crank. Pin the crank to a second fixed point, position it so that the circle the crank forces P to travel passes through O itself. Now invert that circle: a circle through the center of inversion maps to a straight line. Q has no choice — it traces an exact straight line, perpendicular to the line joining the two fixed pivots. Turn the crank and a pen at Q rules a perfectly straight edge.

The math: inversion and the constant product

The cell enforces a single algebraic identity. With long arms of length m (from O to the two side vertices A and C of the rhombus) and rhombus sides of length l, drop a perpendicular from a side vertex onto the O–P–Q axis. A short calculation with the two right triangles gives:

Let A = a side vertex of the rhombus, h = its distance off the OPQ axis,
and x = the foot of the perpendicular measured from O along the axis.

Long arm:    m² = x² + h²            (triangle O–A–foot)
Short bar:   l² = (OP_axis − x)² + h²  with P the near vertex
             l² = (OQ_axis − x)² + h²  with Q the far vertex   ... by symmetry P,Q sit either side of the foot

Subtract and the h² and x² cancel:

   OP × OQ = m² − l²   =  k²   (a constant — the inversion radius squared)

This is the defining equation of an inversion of radius k = √(m² − l²). The map sends a point P at distance r = OP to the point Q at distance k²/r on the same ray. Inversion has one property we exploit:

A circle that passes through the center of inversion O
        maps to a straight line (not through O).

Proof sketch: a circle through O has polar equation r = D·cos(θ − φ),
where D is its diameter. Its inverse r' = k²/r = k² / (D·cos(θ − φ)),
i.e.  r'·cos(θ − φ) = k²/D = constant  —  the polar equation of a line.

So the crank that forces P onto a circle through O is precisely the input that turns Q's locus into a line. The output line stands a fixed distance k²/D from O, where D is the diameter of the crank circle. Drawn out: a perfect straight line generated by nothing but revolute pins and rigid bars — no slides, no rails, no curved templates.

Worked example: sizing a desktop inversor

Suppose you want to build a demonstration model and you have these stock parts: long arms m = 100 mm, rhombus bars l = 60 mm. The inversion constant is fixed by the bars alone:

k² = m² − l² = 100² − 60² = 10000 − 3600 = 6400 mm²
k  = 80 mm   (the inversion radius)

If P sits at OP = 50 mm, then  OQ = k²/OP = 6400/50 = 128 mm.
If P sits at OP = 40 mm, then  OQ = k²/OP = 6400/40 = 160 mm.
Halve the input distance and the output distance doubles — strongly nonlinear.

Now add the crank. To make Q draw a line, the crank circle must pass through O. Pick a crank pivot B placed at distance b from O and a crank length c; the circle passes through O exactly when b = c (the pivot sits one radius away from O, so O is on the circle). Choose b = c = 45 mm. Then the crank circle has diameter D = 2c = 90 mm, and the output line stands off O by:

offset = k² / D = 6400 / 90 ≈ 71.1 mm  from O, perpendicular to the line OB.

The straight segment Q can sweep is bounded by the rhombus reach:
its length scales with the crank's angular travel — a half-turn of the crank
sweeps Q across the usable straight stroke, a few tens of millimetres here.

The straightness is exact in the ideal geometry. In a real model the error budget is set entirely by pin clearance and bar-length tolerance: a 0.05 mm slop at each of six pins, compounded through the inversion's amplification near the crank's O-crossing, is what limits a hand-built model — not any approximation in the math.

Peaucellier vs other straight-line linkages

	Peaucellier-Lipkin	Hart inversor	Watt linkage	Chebyshev linkage	Hoeken / Roberts
Straightness	Exact	Exact	Approximate	Approximate	Approximate
Principle	Circle inversion (rhombus)	Circle inversion (antiparallelogram)	Coupler-curve midpoint	Coupler-curve symmetry	Coupler curve
Number of bars	7	5	3	3	3 to 4
Pin axes (incl. grounded)	6 (2 grounded)	6 (2 grounded)	4	4	4 to 5
Degrees of freedom	1	1	1	1	1
Compactness	Bulky (rhombus + arms)	Compact (bars cross)	Very compact	Compact	Compact
Year	1864 / 1871	1874	1784	~1850s	1860s+
Best for	Teaching exactness, precision guides	Compact exact line	Beam-engine pistons	Cheap near-straight stroke	Long flat strokes, walkers

The dividing line in the table is the top row. Watt, Chebyshev, Hoeken and Roberts linkages are all approximate — they trace a coupler curve that happens to be nearly straight over a useful arc, then bends away. They are simpler (three bars, four pins) and good enough for an engine piston, which only travels a fixed stroke. The Peaucellier and Hart are the only two exact straight-line linkages in common use, and both pay for exactness with extra bars and the geometric machinery of inversion.

Where it came from: solving a century-old problem

James Watt called the approximate straight-line motion he devised in 1784 "one of the most ingenious, simple pieces of mechanism I have contrived." It guided the piston rod of his beam engine along a near-straight path so the rod would not bind — but only near straight, tracing a slender figure-eight (a lemniscate-like curve) that is rectilinear only at its center. For the next eighty years the open question stood: can a linkage of rigid bars and pin joints produce an exact straight line?

Charles-Nicolas Peaucellier, a French army engineer (Génie), published the answer in 1864 — initially as a terse note, almost a teaser, without the full mechanism. Independently, Yom Tov Lipman Lipkin, a Lithuanian-Jewish mathematician working in Russia, built and described the complete linkage around 1871 and was nearly credited as sole inventor before Peaucellier's priority was recognized; the device carries both names. The mathematician James Joseph Sylvester popularized it in England, and the story goes that when Lord Kelvin was shown a working model he refused to let go of it, saying it was "the most beautiful thing I have ever seen in my life."

The linkage's deeper legacy is theoretical. In 1876 A. B. Kempe proved his universality theorem: for any algebraic plane curve, there exists a linkage that traces it — popularly summarized as "you can build a linkage to sign your name." The Peaucellier cell, by mechanically realizing inversion, is a key building block in modern constructive proofs of that theorem.

Where it is used — and why so rarely

Here is the irony that every kinematics lecturer points out: the exact straight-line linkage arrived too late to matter as a machine element. By 1864 the metal-planing machine and the precision-ground slide had already given engineers cheap, stiff, accurate prismatic guides. A sliding way is simpler, stiffer, and takes load better than a delicate seven-bar loop with six pin joints. The straight-line problem was urgent in Watt's era; the straight-line solution showed up after the problem had been engineered around.

• Early ventilating engines. The best-documented practical installation drove the air-pump/ventilation blowers serving the British House of Commons in the 1870s, where the Peaucellier motion guided a piston rod.
• Precision instruments. Where a frictionless, backlash-controllable pin-jointed guide is preferable to a sliding surface that wears and needs lubrication, an inversor can guide a probe or stylus in a straight line without rails.
• Teaching and mechanism synthesis. Its overwhelming use today: it is the canonical example in every theory-of-machines and kinematic-synthesis course, the cleanest physical demonstration of geometric inversion, and a fixture in mathematics museums and "mechanism" model collections (the Reuleaux kinematic model collections, science museums, etc.).
• Conceptual ancestor. Its inversion idea seeds discussion of Hart's inversor, Kempe's universality, and modern linkage-design and origami/robotics folding theory.

Design notes, failure modes, and pitfalls

• The bar-length condition is the whole device. The two long arms must be exactly equal and the four rhombus bars must be exactly equal, or O–P–Q stops being collinear and the inversion identity OP × OQ = m² − l² drifts. Any straightness error traces directly back to a length error, not to wear in service.
• The crank circle must pass through O. If the crank's circle does not pass exactly through the center of inversion, Q traces a circular arc, not a line — because the inverse of a circle not through O is another circle. This is the single most common build mistake: get the crank-pivot distance wrong and you get an arc that looks almost straight but bows.
• Singularities at the ends of travel. When the rhombus flattens out (P, the side vertices, and Q nearly colinear), the mechanism approaches a singular configuration where mobility momentarily degenerates and the linkage can "flip" to the mirror branch. Keep the working stroke away from full flattening.
• Pin clearance amplification. Inversion amplifies motion strongly when OP is small (output ∝ 1/OP). Near that regime, the same pin slop produces a larger output error. Precision models use jeweled or close-fit pivots and stiff, dimensionally-stable bars.
• It carries little load. A seven-bar pin-jointed loop is far more compliant than a sliding way of the same size. Treat it as a motion-defining element, not a load-bearing guide — the reason industry chose slides.
• Misconception: "it approximates a line, like Watt's." No. Within rigid-body geometry the Peaucellier output is mathematically exact. That exactness is the entire point, and what separated it from the century of approximate linkages before it.