Serpentine curve
y = abx/(x² + a²): an S-bend that rises to b/2, falls back and fades to the x-axis on both sides.
Adjust parameters
Export
Formulas and key values
- Cartesian equation
- Maximum
- Inflection points
- Asymptote
a is where the bend peaks and b twice its height.
Explainer video
Make a video of the Serpentine curve
Opens a video template with the parameters on screen. Every number in the video is computed before it renders; choose landscape or vertical.
Start the video01 · Definition
What is the Serpentine curve?
The serpentine is a cubic curve that winds like a snake through the origin. It was studied by l’Hôpital and Huygens in the 1690s, and Isaac Newton gave it its name in his enumeration of cubic curves.
Written as a function, y = abx/(x² + a²), it is easy to analyse: an odd function with a maximum b/2 at x = a, a minimum −b/2 at x = −a, inflection points at x = 0 and x = ±√3·a, and the x-axis as its asymptote in both directions.
02 · Properties
Key properties
- Maximum b/2 at x = a; minimum −b/2 at x = −a.
- Inflection points at x = 0 and x = ±√3·a.
- The x-axis is an asymptote on both sides.
- Odd function: symmetric about the origin.
- Named by Newton in his classification of cubic curves.
03 · Questions
Serpentine curve: common questions
- What is the equation of the Serpentine curve?
- Cartesian equation: y = abx/(x^2 + a^2). a is where the bend peaks and b twice its height.
04 · Keep exploring