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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 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, 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. 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 g ...
and Johann Bernoulli 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 ...
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 ...
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 t ...
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 can ...
: :\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, :\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, 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