Three-bar motion
One aspect of 19th-century research on curves was on those curves that were
generated by mechanical rods. This type of question was an extension of the
problem solved approximately by the Scottish engineer James Watt who
designed jointed rods to turn circular motion into linear motion. In the steam age
this was a significant step forward.
The simplest of these mechanical gadgets is the three-bar motion, where the
bars are jointed together with fixed positions at either end. If the ‘coupler bar’ PQ
moves in any which way, the locus of a point on it turns out to be a curve of
degree six, a ‘sextic curve’.
Algebraic curves
Following Descartes, who revolutionized geometry with the introduction of x,
y and z coordinates and the Cartesian axes named after him, the conics could
now be studied as algebraic equations. For example, the circle of radius 1 has the
equation x^2 + y^2 = 1, which is an equation of the second degree, as all conics
are. A new branch of geometry grew up called algebraic geometry.
In a major study Isaac Newton classified curves described by algebraic
equations of degree three, or cubic curves. Compared with the four basic conics,
78 types were found, grouped into five classes. The explosion of the number of
different types continues for quartic curves, with so many different types that the