Lipkin linkage

Since James Watt invented the steam engine there was a problem of trans­forming rotary motion of one hinge into perfect straight-line motion of another. Thus, a straight-line mech­a­nism.

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During a long time scien­tists and engi­neers couldn't solve this problem, were constructing approx­i­mate solu­tions drawing not perfect straight lines. And only beau­tiful math­e­matics helped finally to solve this problem.

Let's recall that plane circle inver­sion is a bijec­tive mapping of the inner part of the circle (except a single point, the center) onto the outer part. The image of the point AA is the point AA’ lying on the ray starting at the center and passing through AA. Its posi­tion is deter­mined by the equality OAOA=R2OA \cdot OA'=R^2. Inver­sion helps to solve many inter­esting prob­lems in geom­etry. As we'll se, it helps not only in theo­ret­ical ques­tions.

Consider a linkage with one fixed red hinge. The ends of two equal edges are attached to a hinged rhomb.

This mech­a­nism imple­ments inver­sion with respect to a circle centered in the fixed hinge and radius depending on the edge lengths.

Let's see using our mech­a­nism what prop­er­ties inver­sion has.

From the defi­n­i­tion of inver­sion it is clear that the image of a segment of a line passing through the center is mapped again to a segment lying on the same line.

The image of a segment of a line not passing through the center is an arc of a circle, passing through the center of inver­sion.

A circle not passing through the center of inver­sion and not inter­secting the inver­sion circle maps again to a circle.

Inver­sion changes orien­ta­tion. Similar trans­for­ma­tions in math­e­matics are called anti-conformal (conformal are those that preserve both angles and their orien­ta­tion).

An arc of a circe passing through the center of inver­sion maps to… a strict straight-line segment!

Exactly this prop­erty has been used to construct the straight-line linkage. Add a fixed hinge in the center of the circle and an edge of radius length so that the leading hinge follows the circular path. Thus, the driven hinge will always follow a straight-line segment. This kind of mech­a­nisms are often called inver­sors.

In 1864 in a private letter an engi­neer of the French army Charles Nicolas Peaun­cel­lier (1823—1913) announced his construc­tion of an inversor. However, he didn't mention any details. In 1868 a student of P. L. Tcheby­shev, Lipman Lipkin (1846—1876), invents an inversor. His detailed paper was published in 1870 and only in 1873 Peaun­cel­lier publishes his one describing exactly the same mech­a­nisms and citing Lipkin.

After­wards, there were other link­ages constructed based on other math­e­mat­ical ideas. However, the inversor is notable for its beauty, good mechan­ical prop­er­ties and is widely used in tech­niques.