Serpentine Curve
SerpentineCurve
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
| [画像: y=(abx)/(x^2+a^2). ] |
(1)
|
It has parametric equations
x = acott
(2)
y = bsintcost
(3)
for 0<t<pi or
x = atant
(4)
y = bsintcost
(5)
for -pi/2<t<pi/2.
The curve has a maximum at x=a and a minimum at x=-a, where
Interestingly, the minimum and maximum values are +/-b/2, which are independent of a.
And inflection points at x=+/-sqrt(3)a, where
In the parametric representation, the curvature is given by
![画像: kappa(t)=-(2abcott(cot^2t-3))/([b^2cos^2(2t)+a^2csc^4t]^(3/2)).](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/SerpentineCurve/NumberedEquation4.svg){kind=link}
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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." https://mathshistory.st-andrews.ac.uk/Curves/Serpentine/.Smith, D. E. History of Mathematics, Vol. 2: Special Topics of Elementary Mathematics. New York: Dover, p. 330, 1958.Referenced on Wolfram|Alpha
Serpentine CurveCite this as:
Weisstein, Eric W. "Serpentine Curve." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SerpentineCurve.html