Kappa curve

The kappa curve has two vertical asymptotes.

In geometry, the kappa curve or Gutschoven's curve is a two-dimensional algebraic curve resembling the Greek letter κ (kappa).

Using the Cartesian coordinate system it can be expressed as:

x²(x² + y²) = a²y²

or, using parametric equations:

$\begin{matrix} x&=&a\sin t\\ y&=&a\sin t\tan t \end{matrix}$

In polar coordinates its equation is even simpler:

r = atan θ

It has two vertical asymptotes at $x=\pm a$, shown as dashed blue lines in the figure at right.

The kappa curve's curvature:

$\kappa(\theta)={8\left(3-\sin^2\theta\right)\sin^4\theta\over a\left[\sin^2(2\theta)+4\right]^{3\over2}}$

Tangential angle:

$\phi(\theta)=-\arctan\left[{1\over2}\sin(2\theta)\right]$

The kappa curve was first studied by Gérard van Gutschoven around 1662. Other famous mathematicians who have studied it include Isaac Newton and Johann Bernoulli. Its tangents were first calculated by Isaac Barrow in the 17th century.

Derivative

By using implicit differentiation, it is possible to find that the derivative of the kappa curve is:

$2y \frac{dy}{dx}(x^2 + y^2) + y^2(2x + 2y \frac{dy}{dx}) = 2a^2x$

$\frac{dy}{dx}(2yx^2 + 4y^3) + 2xy^2 = 2a^2x$

$\frac{dy}{dx} = \frac{x(a^2 - y^2)}{y(x^2 + 2y^2)}$

External links

• Weisstein, Eric W., "Kappa curve" from MathWorld.
• A Java applet for playing with the curve