Serpentine curve

A serpentine curve is a curve whose equation is of the form

[math]\displaystyle{ x^2y+a^2y-abx=0, \quad ab \gt 0. }[/math]

Equivalently, it has a parametric representation

[math]\displaystyle{ x=a\cot(t) }[/math], [math]\displaystyle{ y=b\sin (t)\cos(t), }[/math]

or functional representation

[math]\displaystyle{ y=\frac{abx}{x^2+a^2}. }[/math]

The curve has an inflection point at the origin. It has local extrema at [math]\displaystyle{ x = \pm a }[/math], with a maximum value of [math]\displaystyle{ y=b/2 }[/math] and a minimum value of [math]\displaystyle{ y=-b/2 }[/math].

History

Serpentine curves were studied by L'Hôpital and Huygens, and named and classified by Newton.

Visual appearance

The serpentine curve for a = b = 1.

External links

• MathWorld – Serpentine Equation
• [1]