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French curves are a set of curvilinear rulers used in industrial design, before the advent of CAD, when everything still had to be drawn by hands.

The most popular set of such rulers is made up of 3 rulers chosen among a set of 28 that were described in a book by Burmester at the beginning of the XXth century. (Lexicon der Gesamten Technik (1904)). For this reason they are often known also as Burmester's curve.

In the engineering US literature they are often mentioned also as irregular curves.

I have the following questions:

  1. Does anyone knows which curves are describing their profiles? Either of the 3 or of the full set of 28.
  2. Connected to the previous: wikipedia (without source) and some others mention sometimes the fact that profile curves contain arcs of clothoid. Though quite reasonable from some points of view (linearly approximating curvature) I've found no explicit evidence for this. Any reference you may know for this "factoid"?
  3. Burmester certainly built up his 28 profiles for some reasons: is there any math reason for the 28 profiles?
  4. Burmester certainly was capable of physically buildings his models: how were their constructed? Since Burmester is known for his work on geometric linkages is there any evidence that such profiles can be described via a (simple?) planar linkage?

Any help on even one of these topics is welcome.

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    $\begingroup$ My guess for the "three curves" is here: hsm.stackexchange.com/a/3638/229 $\endgroup$ – Gerald Edgar Jan 31 '18 at 13:35
  • $\begingroup$ Yes. These are the three profiles I was referring to... $\endgroup$ – Nicola Ciccoli Jan 31 '18 at 14:09
  • $\begingroup$ It is interesting that Lexicon does not call them French or Burmester curves but rather Kurvenlineale, curved rulers, Russian term "lecala" (roughly "benders") is uncharacteristically derived from an ancient Slavic word rather than from a Latin root, as usual for such terminology. I wonder how these tools were introduced in different countries and the names coined. $\endgroup$ – Conifold Feb 1 '18 at 1:24
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There seems to be little secondary literature on this so answering the OP questions fully would take some serious digging into the original sources. One promising secondary source that I was unable to locate is Ceccarelli and Koetsier, Burmester and Allevi: A Theory and its Application for Mechanism Design at the end of 19th Century, Proceedings of IDETC/CIE 2006 ASME 2006 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference. I will only give some comments.

There has been a revival of interest in the old drawing techniques recently due to their digitization, for touch-based input sketch-based techniques give a more natural feel and fluidity than conventional interfaces that employ control point manipulation of splines like Bezier curves. I suspect that the reference to clothoid comes from modern digital reconstructions of French curves, for instance McCrae and Singh stitch them from clothoid pieces. But there is no connection to Burmester's works, only reference to the Lexicon der gesamten Technik, and earlier Singh used cubic NURBS (Non-Uniform Rational B-Splines) to the same end. By the way, Lexicon der gesamten Technik (Encyclopaedia of All Technology and its Auxiliary Sciences) is not a book by Burmester, it is a multi-volume encyclopedia first published by Lueger in 1894–1899 that had a picture of a complete Burmester set, it is unclear that Burmester directly contributed to it.

Burmester's doctoral thesis was on the geometry of isophotes (lines of equal brightness), but his opus magnum Lehrbuch der Kinematik (Textbook of Kinematics, 1888) developed kinematic synthesis of curve-drawing linkages, an elegant and sadly forgotten theory originated by Reuleaux (Gibson has a brief section on it in his Elementary Geometry of Algebraic Curves (1998), which ironically does not even mention Reuleaux). Roughly, the idea was to sketch various curves by designing linkages and other mechanical devices whose points would trace them in motion, such as the classical four-bar linkage employed in Watt's engine. It seems likely that the curves Burmester used were linkage coupler curves (traced by a point on the coupler bar). The cubic of stationary curvature (an asymmetric strophoid, below), traced by the four-bar linkage, is reminiscent of the shapes in French curves. Hartenberg's Kinematic Synthesis of Linkages, published when the subject was still lively (1964), gives some history:

"Modern kinematics had its beginning with Reuleaux. His now classical "Theoretische Kinematik" of 1875 presented many views finding general acceptance then that are current still... Reuleaux regarded a mechanism as a (kinematic) chain of connected links (or parts), one link being fixed... Through hindsight we saw in Watt's straight-line linkage the amorphous beginning of an ordered and advanced synthetic process. Nearly a century later Reuleaux identified synthesis as a concept, as an entity that might be and must be pursued to guide the designer through the maze of mechanisms. His views were limited to only type synthesis: by this is meant the determination of the type of mechanism for a given job... which lies ahead of the related and more recent fields of number and dimensional synthesis.

[...] Geometers and algebraicists of the 1870s became interested in linkages as curve-drawing devices, not as hardware; their work has since been made part of the corpus of the kinematics of mechanisms. It was discovered that a link motion can be found to describe an algebraic curve of any order. In particular, the coupler-point motion of the four-bar linkage (then called a "three-bar motion") was studied. Samuel Roberts showed that a coupler point describes a curve of the sixth order.

[...] Concepts far beyond those of Reuleaux were added to the picture he had attempted to paint; the whole is now called synthesis of linkages... The third and last kind is dimensional synthesis, or the determination of the proportions (lengths) of the links needed to accomplish the specified motion transformation. In Germany, Burmester was in accord with Reuleaux's fundamental concepts and most of his nomenclature and definitions. Making extensive use of mathematical principles (mostly geometrical), and considering displacement, velocity, and acceleration, Burmester's "Lehrbuch der Kinematik" (1888) developed geometric methods that furthered analysis and showed the way to synthesis.

By the way, Hartenberg uses "analysis and synthesis" in their original ancient sense described by Pappus.

Cubic of stationary curvature traced by a four-bar linkage:

enter image description here

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  • $\begingroup$ I read Ceccarelli and Koetsier's paper but, unfortunately, does not add much to what you're saying. I wait a bit to see if I can collect some additional information before accepting your (beautiful though not conclusive) answer. $\endgroup$ – Nicola Ciccoli Feb 1 '18 at 10:19

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