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)
|
and parametric equation s
 |
 |
 |
(2)
|
 |
 |
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(3)
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The curve has a maximum at 
and a minimum at 
, where
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(4)
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Interestingly, the minimum and maximum values are 
, which are independent of 
.
And inflection points at 
, where
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(5)
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In the parametric representation, the curvature is given by
![kappa(t)=-(2abcott(cot^2t-3))/([b^2cos^2(2t)+a^2csc^4t]^(3/2)).](http://mathworld.wolfram.com/images/equations/SerpentineCurve/NumberedEquation4.gif) |
(6)
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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.
