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:
- Failed to parse (Missing texvc executable; please see math/README to configure.): x^2(x^2 + y^2)=a^2y^2
or, using parametric equations:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{matrix} x&=&a\sin t\\ y&=&a\sin t\tan t \end{matrix}
In polar coordinates its equation is even simpler:
- Failed to parse (Missing texvc executable; please see math/README to configure.): r=a\tan\theta
It has two vertical asymptotes at Failed to parse (Missing texvc executable; please see math/README to configure.): x=\pm a , shown as dashed blue lines in the figure at right.
The kappa curve's curvature:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \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:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \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 [link]
By using implicit differentiation, it is possible to find that the derivative of the kappa curve is:
Failed to parse (Missing texvc executable; please see math/README to configure.): 2y \frac{dy}{dx}(x^2 + y^2) + y^2(2x + 2y \frac{dy}{dx}) = 2a^2x
Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{dy}{dx}(2yx^2 + 4y^3) + 2xy^2 = 2a^2x
Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{dy}{dx} = \frac{x(a^2 - y^2)}{y(x^2 + 2y^2)}
