Kappa curve

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Kappa_curve_with_asymptotes_-_by_Pt.png

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:

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 G�rard 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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