Serpentine Curve

Calculus Curves >

The serpentine curve, named because of its snakelike shape, is a cubic curve defined by the Cartesian equation [1]
serpentine curve rectangular equation

or by the parametric equations
serpentine curve parametric equation

The serpentine is a subcase of one of the normal forms of the cubic curve f(x, y) = 0:
xy2 + ey = ax3 + bx2 + cx + d [1].

graph of serpentine curve
Graph of a serpentine curve.

Serpentine Curve Properties


The serpentine curve was first studied by L’Hôpital and Huygens in 1692. Later, it was studied by Newton, in 1701, who gave the curve its name [3]. It appears in the 1710 work Curves by Sir Isaac Newton in Lexicon Technicumby John Harris [1].

Applications of the Serpentine Curve

Snakes, perhaps not surprisingly, move in the pattern of a serpentine curve. According the Hirose [4], that’s because the curve has the “greatest amount of smoothness of contraction and relaxation of the motor muscles”. The model for snake movement is a little more challenging than the generic serpentine curve proposed by Newton:
snakes serpentine curve


Serpentine curves are also used in surveying, where they are also called S-curves. They are generally used to connect two railway lines or parallel roads intersect at a tiny angle [5].

In engineering, the trajectory of a Chaplygin sleigh with periodic actuation is a serpentine curve [6].


[1] Mactutor. Serpentine. Retrieved March 6, 2022 from:
[2] Weisstein, Eric W. “Serpentine Curve.” From MathWorld–A Wolfram Web Resource.
[3] Titu Maiorescu University. The International Conference Education and Creativity for a Knowledge based Society – Computer Science, 2012.
[4] S. Hirose. Biologically Inspired Robots: Snake-Like Locomotors and
Manipulators. Oxford University Press, 1993. Cited in Spranklin, B. DESIGN, ANALYSIS, AND FABRICATION OF A SNAKE-INSPIRED ROBOT WITH A RECTILINEAR GAIT. 2006.
[5] Types of Curves in Surveying. Retrieved March 6, 2022 from:
[6] Fedonyuk, V. (2020). Dynamics and Control of Nonholonomic Systems with Internal Degrees of Freedom. Retrieved March 6, 2022 from:

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