geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

, the kappa curve or Gutschoven's curve is a two-dimensional

algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane ...

resembling the

Greek letter The Greek alphabet has been used to write the Greek language since the late 9th or early 8th century BCE. It is derived from the earlier Phoenician alphabet, and was the earliest known alphabetic script to have distinct letters for vowels as ...

. The kappa curve was first studied by Gérard van Gutschoven around 1662. In the

history of mathematics The history of mathematics deals with the origin of discoveries in mathematics and the mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments ...

, it is remembered as one of the first examples of Isaac Barrow's application of rudimentary calculus methods to determine the

tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...

of a curve.

Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a " natural philosopher"), widely recognised as one of the grea ...

and

Johann Bernoulli Johann Bernoulli (also known as Jean or John; – 1 January 1748) was a Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is known for his contributions to infinitesimal calculus and educating Le ...

continued the studies of this curve subsequently. Using the Cartesian coordinate system it can be expressed as :

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

or, using parametric equations, :

\begin
x &= a\sin t,\\
y &= a\sin t\tan t.
\end

In polar coordinates its equation is even simpler: :

r = a\tan\theta.

It has two vertical

asymptote In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related context ...

s at , shown as dashed blue lines in the figure at right. The kappa curve's curvature: :

\kappa(\theta) = \frac.

Tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...

ial angle: :

\phi(\theta) = -\arctan\left(\tfrac12 \sin(2\theta)\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\left(x^2 + y^2\right)-a^2y^2 = 0

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

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

Distributing the differential and applying appropriate rules, :

\frac &= \frac \end

Derivative

If we use the modern concept of a functional relationship and apply

implicit differentiation In mathematics, an implicit equation is a relation of the form R(x_1, \dots, x_n) = 0, where is a function of several variables (often a polynomial). For example, the implicit equation of the unit circle is x^2 + y^2 - 1 = 0. An implicit func ...

, the slope of a tangent line to the kappa curve at a point is: :

\frac &= \frac \end

References

External links

*
A Java applet for playing with the curve
*{{MacTutor, class=Curves, id=Kappa, title=Kappa Curve Plane curves