A French curve is a template usually made from metal, wood or plastic composed of many different curves. It is used in manual drafting to draw smooth curves of varying radii. The shapes are segments of the Euler spiral or clothoid curve. The curve is placed on the drawing material, and a pencil, knife or other implement is traced around its curves to produce the desired result.¹

A complete Burmester set from the Lexikon der gesamten Technik (1904)

The outlines of these tools were probably shaped by tracing flexible fibrous wood sticks, or by following the soft lead that also enables the construction of lesbian rules².

How is it that these aids were established? By who and how were the master templates for these curves designed? Were these also produced by trusting stressed strips of wood, and so on?

Using a French curve

In the very existence of the French Curve, we find the acknowledgement that somehow freehand curve drawing is not sufficient, it is not mastered.

We looked at the mathematical theory of the four-bar linkage that kinematician and geometer Ludwig Burmester³ introduced in the late 19th century, these geometric techniques for the synthesis of linkages were served.

to compute the geometric constraints of [...] linkage directly from the inventor's desired movement for a floating link. From this point of view a four-bar linkage is a floating link that has two points constrained to lie on two circles⁴.

Plotting curves with a four-bar linkage.

The four-bar linkage describes a world of rods, cranks and sliders. It opens up another way to parametrise a naturally flowing curvature, one that does not rely on flexible materials but requires a balanced adjustment of control points for it to function.

Would there be a way to extract some of the parameters used in four-bar linkages, for drawing curved?

Inverse kinematics of SCARA robot done with MeKin2D, a software used for the kinematic simulation of planar linkages, for synthesis and analysis of disk-cam mechanisms, and for involute gear generation.