Kappa curve

 ( Quartic Curves ) 

In geometry, the kappa curve or Gutschoven's curve is a two-dimensional algebraic curve resembling the Greek alphabet . 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.

---

Formulation Using the Cartesian coordinate system it can be expressed as

$x^2\backslash left(x^2\; +\; y^2\backslash right)\; =\; a^2y^2$

or, using parametric equations,

$\backslash begin\{align\}$

x &= a\sin t,\\ y &= a\sin t\tan t. \end{align}

In polar coordinates its equation is even simpler:

$r\; =\; a\backslash tan\backslash theta.$

It has two vertical at , shown as dashed blue lines in the figure at right.

The kappa curve's curvature:

$\backslash kappa(\backslash theta)\; =\; \backslash frac\{8\backslash left(3\; -\; \backslash sin^2\backslash theta\backslash right)\backslash sin^4\backslash theta\}\{a\; \backslash left(\backslash sin^2(2\backslash theta)\; +\; 4\backslash right)^\backslash frac32\}.$

angle:

$\backslash phi(\backslash theta)\; =\; -\backslash arctan\backslash left(\backslash tfrac12\; \backslash sin(2\backslash theta)\backslash right).$

---

Tangents via infinitesimals The tangent lines of the kappa curve can also be determined geometrically using differentials and the elementary rules of infinitesimal arithmetic. Suppose and are variables, while a is taken to be a constant. From the definition of the kappa curve,

$x^2\backslash left(x^2\; +\; y^2\backslash right)-a^2y^2\; =\; 0$

Now, an infinitesimal change in our location must also change the value of the left hand side, so

$d\; \backslash left(x^2\backslash left(x^2\; +\; y^2\backslash right)-a^2y^2\backslash right)\; =\; 0$

Distributing the differential and applying appropriate rules,

$\backslash begin\{align\}$

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}

---

Derivative If we use the modern concept of a functional relationship and apply implicit differentiation, the slope of a tangent line to the kappa curve at a point is:

$\backslash begin\{align\}$

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}

• J. Dennis Lawrence (1972). 9780486602882, Dover. . ISBN 9780486602882

---

External links

• A Java applet for playing with the curve

Post Comment