Kappa Curve
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In
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 c ...
, 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 c ...
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 w ...
. The kappa curve was first studied by
Gérard van Gutschoven Gérard (French language, French: ) is a French masculine given name and surname of Germanic languages, Germanic origin, variations of which exist in many Germanic and Romance languages. Like many other Germanic name, early Germanic names, it is ...
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 Isaac Barrow (October 1630 – 4 May 1677) was an English Christian theologian and mathematician who is generally given credit for his early role in the development of infinitesimal calculus; in particular, for proof of the fundamental theorem ...
'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. More ...
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 L ...
continued the studies of this curve subsequently. Using the
Cartesian coordinate system A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
it can be expressed as :x^2\left(x^2 + y^2\right) = a^2y^2 or, using
parametric equation In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric obj ...
s, :\begin x &= a\sin t,\\ y &= a\sin t\tan t. \end In
polar coordinates In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the or ...
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 contexts, ...
s at , shown as dashed blue lines in the figure at right. The kappa curve's
curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the canonic ...
: :\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. More ...
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 In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referr ...
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, :\begin d \left(x^2\left(x^2 + y^2\right)\right)-d \left(a^2y^2\right) &= 0 \\ px(2 x\,dx ) \left(x^2+y^2\right) + x^2 (2x\,dx + 2y\,dy) - a^2 2y\,dy &= 0 \\ px\left( 4 x^3 + 2 x y^2\right) dx + \left( 2 y x^2 - 2 a^2 y \right) dy &= 0 \\ pxx \left( 2 x^2 + y^2 \right) dx + y \left(x^2 - a^2\right) dy &= 0 \\ px\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 functi ...
, the slope of a tangent line to the kappa curve at a point is: :\begin 2 x \left( x^2 + y^2 \right) + x^2 \left( 2x + 2 y \frac \right) &= 2 a^2 y \frac \\ px2 x^3 + 2 x y^2 + 2 x^3 &= 2 a^2 y \frac - 2 x^2 y \frac \\ px4 x^3 + 2 x y^2 &= \left(2 a^2 y - 2 x^2 y \right) \frac \\ px\frac &= \frac \end


References

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External links

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