Machine Kinematics
solved, then drawn
Field

The paths points trace

A point on a coupler draws a curve of degree six. Choosing the linkage that draws the curve you want is the oldest hard problem in the subject, and straight lines were the hardest of all.
ground 4, crank 1, coupler 3.5, rocker 35 attachment points, 150 solves each

What a coupler point draws

A point rigidly attached to the coupler of a four-bar traces a curve of degree six. Move the attachment a little and the curve changes a great deal. For most of the twentieth century the practical way to find the linkage that draws a wanted curve was to look it up in a book of printed atlases.

 10⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹0%25%50%75%100%Watt, 1784Chebyshev, 1850sPeaucellier, 1864fraction of the available stroke useddeviation ÷ span120 solved positions per pointfifteen decades, and one flat line

The straight-line problem

Before 1800 a long true flat surface was harder to make than almost anything else, so guiding a piston straight without a slide was worth solving. Watt's answer was an approximation. Measuring how good an approximation, over how much of the stroke, turns out to be a more interesting question than whether it is exact.

 PQOCarm 5, rhombus 3, crank 1.6deviation 6.5e-16 of span

Peaucellier and the exact answer

Eighty years after Watt settled for an approximation, a French army officer found a linkage that draws an exactly straight line from pin joints alone. It works by inversion in a circle, the product it holds constant is measurable, and on this site it comes out straight to 10⁻¹⁶ of its span.

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