Applications of projective geometry The Peaucellier–Lipkin linkage illustrates practical use of an apparently theoretical fact concerning inversions in cicles. Are there applications of this type of theorems of projective geometry?
At wiki we find

During the Industrial Revolution, mechanisms for converting rotary into linear motion were widely adopted in industrial and mining machinery, locomotives and metering devices. Such devices had to combine engineering simplicity with a high degree of accuracy, and the ability to operate at speed for lengthy periods. For many purposes approximate linear motion is an acceptable substitute for exact linear motion. Perhaps the best known example is the Watt four bar linkage, invented by the Scottish engineer James Watt in 1784.

This appears to imply that precise linear linkages (like the  Peaucellier–Lipkin) were also used, in addition to the approximate ones like Watt's. Which linear linkages were used?
Note.  If anyone is familiar with the remarkable book by Artobolevski
(see this answer) available in Russian, French, and English, I would appreciate some input whether any of the thousands of devices there exploit theorems in projective geometry.
 A: I will not address the "projective geometry" aspect, because I have no elements for that ; furthermore, I don't think it is an adequate framework for studying mechanisms in general.
What I would like to say is that converting linear motion into circular motion can be done by other mechanisms. Here is one which is probably not well known.
The example I am going to give (see Figure) comes from an amazing set of 7 books, translated from Russian 


*

*into French named "Les mécanismes dans la technique moderne" by I. Artobolevski (Editions de Moscou, 1975) that give thousands (!) of mecanisms, some of them very clever. 

*into English under the title "Mechanisms in Modern Engineering Design (Mir Editor, 1977)" by I. Artobolevsky (with a "y"). I am indebted to @dxiv who  pointed out the existence of this translation. Vol. 1, 3 and 4 are freely downlable :
(https://archive.org/details/ArtobolevskyMechanismsInModernEngineeringDesignVol1)
(https://archive.org/details/ArtobolevskyMechanismsInModernEngineeringDesignVol3)
(https://archive.org/details/ArtobolevskyMechanismsInModernEngineeringDesignVol4)
Here is mechanism #1770 (Vol. 2 p. 96) converting rotation into translation:

(My translation from French into English) : 
$AB=BC=r$, $R/r=2$; $EG=FH$, $GK=HL$, $EG=GH=KL$. 
The mechanism is based on Cardan circles, i.e., cogwheel #$2$, with radius $r$ rolls on cogwheel #$3$ with radius $R$. Element $1$ generates rotation torques $A$ and $B$ with respect to the fixed element and with wheel #$2$ resp. On observing relationship between the radii of #$2$ and #$3$, point $C$ belonging to wheel #$2$ moves along straight line p--p passing through point $A$. Element $4$ of the translator, which constitutes $2$ parallelogram linkage $KGHL$ and $GEFH$, generates a rotation  torque $C$  with wheel #$2$. When element $1$ turns about fixed axis $A$, element $4$ undergoes a translation move, whereas axis $EF$ of element #$4$ slides along straight line q--q, parallel to direction $KL$. Element $8$ turns about fixed axis $K$, and element $9$ about fixed axis $L$.
A: Here is what I have gathered from the literature on this subject: While Watt's linkage (unlike the Peaucellier–Lipkin linkage, PLL) provides an approximate solution to the problem of converting linear motion to a circular motion, it it the one which is used in practice due to the following reasons:


*

*Development of better synthetic oils (in the late 19th century) to smooth out the motion of Watt's linkage.

*Watt's linkage consists of 4 rods (or 3, depending on what you count as a rod), while PLL consists of 7 rods. What is gained by the mathematical precision of PLL is lost due to slightly higher complexity (imperfection in each extra joint and rod contributes to accumulation of the error in the motion of the "output" joint of the linkage).

*Lack of space in the engines where Watt's linkage is used: Every extra rod requires extra space. 
One can find plenty of toy mechanisms (check youtube) which use PLL, but, in practice, the only use of PLL I am aware of goes back to the late 19th century, when PLL was used for ventilation in the new building of the British Parliament. 
