A curve named and studied by Newton in 1701 and contained in his classification of cubic curves . It had been studied earlier by L'Hospital
and Huygens in 1692 (MacTutor Archive).

The curve is given by the Cartesian equation

(1)

It has parametric equations

for
or

for .

The curve has a maximum at and a minimum at , where

(6)

Interestingly, the minimum and maximum values are , which are independent of .

And inflection points at , where

(7)

In the parametric representation, the curvature is
given by

(8)

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References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 225,
1987. Lawrence, J. D. A
Catalog of Special Plane Curves. New York: Dover, pp. 111-112, 1972. MacTutor
History of Mathematics Archive. "Serpentine." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Serpentine.html . Smith,
D. E. History
of Mathematics, Vol. 2: Special Topics of Elementary Mathematics. New
York: Dover, p. 330, 1958.
Cite this as:
Weisstein, Eric W. "Serpentine Curve."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/SerpentineCurve.html

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