Editing Kappa curve

[[Image:Kappa curve with asymptotes - by Pt.png|frame|The kappa curve has two vertical [[asymptote]]s]]
In [[geometry]], the '''kappa curve''' or '''Gutschoven's curve''' is a two-dimensional [[algebraic curve]] resembling the [[Greek alphabet|Greek letter]] {{not a typo|[[Kappa (letter)|{{math|ϰ}} (kappa)]]}}. The kappa curve was first studied by [[Gérard van Gutschoven]] around 1662. In the [[history of mathematics]], it is remembered as one of the first examples of [[Isaac Barrow]]'s application of rudimentary calculus methods to determine the [[tangent]] of a curve.   [[Isaac Newton]] and [[Johann Bernoulli]] continued the studies of this curve subsequently.

Using the [[Cartesian coordinate system]] it can be expressed as
:<math>x^2\left(x^2 + y^2\right) = a^2y^2</math>
or, using [[parametric equation]]s,
:<math>\begin{align}
x &= a\sin t,\\
y &= a\sin t\tan t.
\end{align}</math>

In [[polar coordinates]] its equation is even simpler:
:<math>r = a\tan\theta.</math>

It has two vertical [[asymptote]]s at {{math|''x'' {{=}} ±''a''}}, shown as dashed blue lines in the figure at right.

The kappa curve's [[curvature]]:
:<math>\kappa(\theta) = \frac{8\left(3 - \sin^2\theta\right)\sin^4\theta}{a \left(\sin^2(2\theta) + 4\right)^\frac32}.</math>

[[Tangent]]ial angle:
:<math>\phi(\theta) = -\arctan\left(\tfrac12 \sin(2\theta)\right).</math>

==Tangents via infinitesimals==

The tangent lines of the kappa curve can also be determined geometrically using [[Differential (infinitesimal)|differentials]] and the elementary rules of [[infinitesimal]] arithmetic. Suppose {{mvar|x}} and {{mvar|y}} are variables, while a is taken to be a constant. From the definition of the kappa curve,
:<math> x^2\left(x^2 + y^2\right)-a^2y^2 = 0 </math>

Now, an infinitesimal change in our location must also change the value of the left hand side, so
:<math>d \left(x^2\left(x^2 + y^2\right)-a^2y^2\right) = 0 </math>

Distributing the differential and applying [[Differential (calculus)|appropriate rules]],

:<math>\begin{align}
d \left(x^2\left(x^2 + y^2\right)\right)-d \left(a^2y^2\right) &= 0 \\[6px]
(2 x\,dx ) \left(x^2+y^2\right) + x^2 (2x\,dx + 2y\,dy) - a^2 2y\,dy &= 0 \\[6px]
\left( 4 x^3 + 2 x y^2\right) dx + \left( 2 y x^2 - 2 a^2 y \right) dy &= 0 \\[6px]
x \left( 2 x^2 + y^2 \right) dx  + y \left(x^2 - a^2\right) dy &= 0 \\[6px]
\frac{ x \left( 2 x^2 + y^2 \right) }{ y \left(a^2 - x^2\right)} &= \frac{dy}{dx} 
\end{align}</math>

==Derivative==

If we use the modern concept of a functional relationship {{math|''y''(''x'')}} and apply [[implicit differentiation]], the slope of a tangent line to the kappa curve at a point {{math|(''x'',''y'')}} is:

:<math>\begin{align}
2 x \left( x^2 + y^2 \right) + x^2 \left( 2x + 2 y \frac{dy}{dx} \right) &= 2 a^2 y \frac{dy}{dx} \\[6px]
2 x^3 + 2 x y^2 + 2 x^3 &= 2 a^2 y \frac{dy}{dx} - 2 x^2 y \frac{dy}{dx} \\[6px]
4 x^3 + 2 x y^2 &= \left(2 a^2 y - 2 x^2 y \right) \frac{dy}{dx} \\[6px]
\frac{2 x^3 + x y^2}{a^2 y - x^2 y} &= \frac{dy}{dx} 
\end{align}</math>

==References==
*{{Cite book|last=Lawrence|first=J. Dennis|title=A Catalog of Special Plane Curves|year=1972|publisher=Dover|location=New York|url=https://archive.org/details/catalogofspecial00lawr/page/n5/mode/2up|url-access=registration|pages=[https://archive.org/details/catalogofspecial00lawr/page/138/mode/2up 139&ndash;141]|isbn=0-486-60288-5}}

==External links==
*{{MathWorld|title=Kappa curve|urlname=KappaCurve}}
*[http://www-groups.dcs.st-and.ac.uk/~history/Java/Kappa.html A Java applet for playing with the curve]
*{{MacTutor|class=Curves|id=Kappa|title=Kappa Curve}}

[[Category:Quartic curves]]