
The Peaucellier–Lipkin linkage (or Peaucellier–Lipkin cell, or Peaucellier–Lipkin inversor), invented in 1864, was the first true planar
straight line mechanism
A straight-line mechanism is a mechanism that converts any type of rotary or angular motion to perfect or near-perfect straight-line motion, or ''vice versa''. Straight-line motion is linear motion of definite length or "stroke", every forwa ...
– the first planar
linkage capable of transforming
rotary motion
Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an ''axis of rotation''. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersec ...
into perfect
straight-line motion, and vice versa. It is named after
Charles-Nicolas Peaucellier (1832–1913), a French army officer, and
Yom Tov Lipman Lipkin (1846–1876), a
Lithuanian Jew and son of the famed Rabbi
Israel Salanter.
Until this invention, no planar method existed of converting exact straight-line motion to circular motion, without reference guideways. In 1864, all power came from
steam engine
A steam engine is a heat engine that performs Work (physics), mechanical work using steam as its working fluid. The steam engine uses the force produced by steam pressure to push a piston back and forth inside a Cylinder (locomotive), cyl ...
s, which had a
piston
A piston is a component of reciprocating engines, reciprocating pumps, gas compressors, hydraulic cylinders and pneumatic cylinders, among other similar mechanisms. It is the moving component that is contained by a cylinder (engine), cylinder a ...
moving in a straight-line up and down a cylinder. This piston needed to keep a good seal with the cylinder in order to retain the driving medium, and not lose energy efficiency due to leaks. The piston does this by remaining perpendicular to the axis of the cylinder, retaining its straight-line motion. Converting the straight-line motion of the piston into circular motion was of critical importance. Most, if not all, applications of these steam engines, were rotary.
The mathematics of the Peaucellier–Lipkin linkage is directly related to the
inversion of a circle.
Earlier Sarrus linkage
There is an earlier straight-line mechanism, whose history is not well known, called the
Sarrus linkage
The Sarrus linkage, invented in 1853 by Pierre Frédéric Sarrus, is a mechanical linkage to convert a limited circular motion to a linear motion or vice versa without reference guideways. It is a spatial six-bar linkage (6R) with two groups ...
. This linkage predates the Peaucellier–Lipkin linkage by 11 years and consists of a series of hinged rectangular plates, two of which remain parallel but can be moved normally to each other. Sarrus' linkage is of a three-dimensional class sometimes known as a
space crank, unlike the Peaucellier–Lipkin linkage which is a planar mechanism.
Geometry

In the geometric diagram of the apparatus, six bars of fixed length can be seen: , , , , , . The length of is equal to the length of , and the lengths of , , , and are all equal forming a
rhombus
In plane Euclidean geometry, a rhombus (: rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhom ...
. Also, point is fixed. Then, if point is constrained to move along a circle (for example, by attaching it to a bar with a length halfway between and ; path shown in red) which passes through , then point will necessarily have to move along a straight line (shown in blue). In contrast, if point were constrained to move along a line (not passing through ), then point would necessarily have to move along a circle (passing through ).
Many different over-all proportions of this linkage are possible. Since points , , must be collinear at all points in the linkage's motion, and countless arm length combinations are viable, then mirror symmetry across isn't necessary. With staying collinear, the only requirement to achieve the intended straight-line motion of are that , that , and for to be constrained to a circular path which crosses . Otherwise, there is no fixed relationship between the lengths of the sides of the figure, the radius of the constraining circular path of , and the lengths of or .
Mathematical proof of concept
Collinearity
First, it must be proven that points , , are
collinear
In geometry, collinearity of a set of Point (geometry), points is the property of their lying on a single Line (geometry), line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, t ...
. This may be easily seen by observing that the linkage is mirror-symmetric about line , so point must fall on that line.
More formally, triangles and are congruent because side is congruent to itself, side is congruent to side , and side is congruent to side . Therefore, angles and are equal.
Next, triangles and are congruent, since sides and are congruent, side is congruent to itself, and sides and are congruent. Therefore, angles and are equal.
Finally, because they form a complete circle, we have
:
but, due to the congruences, and , thus
:
therefore points , , and are collinear.
Inverse points
Let point be the intersection of lines and . Then, since is a
rhombus
In plane Euclidean geometry, a rhombus (: rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhom ...
, is the
midpoint
In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment.
Formula
The midpoint of a segment in ''n''-dim ...
of both line segments and . Therefore, length = length .
Triangle is congruent to triangle , because side is congruent to side , side is congruent to itself, and side is congruent to side . Therefore, angle = angle . But since , then , , and .
Let:
:
Then:
:
:
(due to the Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...
)
:
(same expression expanded)
:
(Pythagorean theorem)
:
Since and are both fixed lengths, then the product of and is a constant:
:
and since points , , are collinear, then is the inverse of with respect to the circle with center and radius .
Inversive geometry
Thus, by the properties of
inversive geometry
In geometry, inversive geometry is the study of ''inversion'', a transformation of the Euclidean plane that maps circles or lines to other circles or lines and that preserves the angles between crossing curves. Many difficult problems in geometry ...
, since the figure traced by point is the inverse of the figure traced by point , if traces a circle passing through the center of inversion , then is constrained to trace a straight line. But if traces a straight line not passing through , then must trace an arc of a circle passing through . ''
Q.E.D.
Q.E.D. or QED is an initialism of the List of Latin phrases (full), Latin phrase , meaning "that which was to be demonstrated". Literally, it states "what was to be shown". Traditionally, the abbreviation is placed at the end of Mathematical proof ...
''
A typical driver

Peaucellier–Lipkin linkages (PLLs) may have several inversions. A typical example is shown in the opposite figure, in which a rocker-slider four-bar serves as the input driver. To be precise, the slider acts as the input, which in turn drives the right grounded link of the PLL, thus driving the entire PLL.
Historical notes
Sylvester
Sylvester or Silvester is a name derived from the Latin adjective ''silvestris'' meaning "wooded" or "wild", which derives from the noun ''silva'' meaning "woodland". Classical Latin spells this with ''i''. In Classical Latin, ''y'' represented a ...
(''Collected Works'', Vol. 3, Paper 2) writes that when he showed a model to
Kelvin
The kelvin (symbol: K) is the base unit for temperature in the International System of Units (SI). The Kelvin scale is an absolute temperature scale that starts at the lowest possible temperature (absolute zero), taken to be 0 K. By de ...
, he “nursed it as if it had been his own child, and when a motion was made to relieve him of it, replied ‘No! I have not had nearly enough of it—it is the most beautiful thing I have ever seen in my life.’”
Cultural references
A monumental-scale sculpture implementing the linkage in illuminated struts is on permanent exhibition in
Eindhoven, Netherlands
Eindhoven ( ; ) is a city and municipality of the Netherlands, located in the southern province of North Brabant, of which it is the largest municipality, and is also located in the Dutch part of the natural region the Campine. With a population ...
. The artwork measures , weighs , and can be operated from a
control panel accessible to the general public.
See also
*
Linkage (mechanical)
A mechanical linkage is an assembly of systems connected so as to manage forces and Motion, movement. The movement of a body, or link, is studied using geometry so the link is considered to be Rigid body, rigid. The connections between links ...
*
Straight line mechanism
A straight-line mechanism is a mechanism that converts any type of rotary or angular motion to perfect or near-perfect straight-line motion, or ''vice versa''. Straight-line motion is linear motion of definite length or "stroke", every forwa ...
References
Bibliography
*
* — proof and discussion of Peaucellier–Lipkin linkage, mathematical and real-world mechanical models
* (and references cited therein)
* Hartenberg, R.S. & J. Denavit (1964
Kinematic synthesis of linkages pp 181–5, New York: McGraw–Hill, weblink from
Cornell University
Cornell University is a Private university, private Ivy League research university based in Ithaca, New York, United States. The university was co-founded by American philanthropist Ezra Cornell and historian and educator Andrew Dickson W ...
.
*
*
External links
How to Draw a Straight Line, online video clips of linkages with interactive applets.How to Draw a Straight Line, historical discussion of linkage designJewish Encyclopedia article on Lippman Lipkinand his father
Israel SalanterPeaucellier Apparatusfeatures an interactive applet
using the Molecular Workbench software
called Hart's Inversor.
Modified Peaucellier robotic arm linkage (Vex Team 1508 video)
{{DEFAULTSORT:Peaucellier-Lipkin Linkage
Linkages (mechanical)
Articles containing proofs
Linear motion
Straight line mechanisms