The Peaucellier-Lipkin linkage transforms a circular arc into a straight path through geometric inversion. A rhombus of equal links and two longer links keep the product of two distances constant. Explore the motion and calculate its line offset and ideal stroke.
Peaucellier-Lipkin Exact-Closure Explorer
Set the two cell-link lengths, the driver-circle radius and angular sweep. Every bar remains rigid while inverse point Q traces the exact output line.
Equation Used
- The two links from fixed center O to opposite cell pivots are equal length a.
- The four rhombus links are equal length b, with a greater than b.
- The driver link CP and fixed spacing OC both equal c.
- All links are ideal rigid planar members with perfect pivots.
- Force, stress, clearance, tolerance, friction, speed, interference and safety are excluded.
The input point moves on a circle through the inversion center, giving a straight path for the output point. Results assume ideal links and pivots; folded positions where P and Q coincide need a separate singularity check.
Eight rigid links form an exact geometric inversor
Cornell's Reuleaux collection identifies the Peaucellier-Lipkin mechanism as an eight-link exact straight-line linkage. Four equal links form the rhombus cell, and two equal longer links connect its side pivots to fixed inversion center O.
The cell enforces OP times OQ equals a squared minus b squared. Constraining P to a circle through O therefore constrains inverse point Q to one exact straight line.
Historical exact straight-line motion
Cornell records applications in steam-engine pressure indicators and machine tools. The linkage is historically important because it produces mathematically exact straight-line motion from revolute joints.
This explorer demonstrates ideal geometry; it does not size an actuator, machine joint or loaded mechanism.
Inversion constant and line position
For a driver circle of radius c whose center is c from O, OP squared equals 2c times the horizontal coordinate of P. Inversion sends P to Q=kP/OP squared, so Q has constant horizontal coordinate k/(2c).
Declared exact example
For a=280 mm, b=200 mm and c=140 mm, k=38,400 mm squared and the exact output line is 137.143 mm from O. A plus-or-minus 65-degree driver swing gives 174.74 mm of ideal straight-line output stroke.
Exact ideal theorem, separate real tolerances
The mathematical result is exact only for rigid links of exact length and perfect planar pivots. A real design needs a configuration-dependent tolerance, stiffness, bearing, interference and load analysis.
Calculating departure from a straight line requires the actual link and pivot errors and their effect through the stroke. This explorer reports ideal geometric results and does not check singularity avoidance.
Peaucellier-Lipkin questions
Why is the output exact?
The linkage makes Q the geometric inverse of P, and inversion maps a circle through O to a straight line.
What stays constant?
OP times OQ equals a squared minus b squared.
Do any bars stretch?
No. All six cell links and the circular driver remain rigid in the animation.
Does exact theory predict manufacturing error?
No. Real error requires a separate tolerance and compliance model.
Peaucellier-Lipkin references
- Cornell University Library, Reuleaux Peaucellier Straight-line Mechanism: eight-link exact straight-line linkage, geometric inversion and historical applications.
- Kinematic Synthesis of Linkages, section 6-6: derivation of the constant product and exact straight-line construction.
- Cornell paper on Reuleaux models and mathematics: circle-through-center inversion explanation.
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