math lessons - Kappa curve

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:

x 4 + x 2 y 2 = a 2 y 2

$\begin{matrix} x&=&a\cos t\,\cot t\\ y&=&a\cos t \end{matrix}$

In polar coordinates its equation is even simpler:

r = a cotθ

It has two vertical asymptotes at $x=\pm a,$ they have been denoted as blue dashed lines on the graphic.

The kappa curve's curvature:

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

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

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.

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