In geometry, the kappa curve or Gutschoven's curve is a two-dimensional algebraic curve resembling the Greek letterϰ (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.
The tangent lines of the kappa curve can also be determined geometrically using differentials and the elementary rules of infinitesimal arithmetic. Suppose x and y are variables, while a is taken to be a constant. From the definition of the kappa curve,
Now, an infinitesimal change in our location must also change the value of the left hand side, so
If we use the modern concept of a functional relationship y(x) and apply implicit differentiation, the slope of a tangent line to the kappa curve at a point (x,y) is:
Referencesedit
Lawrence, J. Dennis (1972). A Catalog of Special Plane Curves. New York: Dover. pp. 139–141. ISBN0-486-60288-5.
kappa, curve, geometry, kappa, curve, gutschoven, curve, dimensional, algebraic, curve, resembling, greek, letter, kappa, kappa, curve, first, studied, gérard, gutschoven, around, 1662, history, mathematics, remembered, first, examples, isaac, barrow, applicat. In geometry the kappa curve or Gutschoven s curve is a two dimensional algebraic curve resembling the Greek letter ϰ kappa The kappa curve was first studied by Gerard 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 The kappa curve has two vertical asymptotes Using the Cartesian coordinate system it can be expressed as x 2 x 2 y 2 a 2 y 2 displaystyle x 2 left x 2 y 2 right a 2 y 2 or using parametric equations x a sin t y a sin t tan t displaystyle begin aligned x amp a sin t y amp a sin t tan t end aligned In polar coordinates its equation is even simpler r a tan 8 displaystyle r a tan theta It has two vertical asymptotes at x a shown as dashed blue lines in the figure at right The kappa curve s curvature k 8 8 3 sin 2 8 sin 4 8 a sin 2 2 8 4 3 2 displaystyle kappa theta frac 8 left 3 sin 2 theta right sin 4 theta a left sin 2 2 theta 4 right frac 3 2 Tangential angle ϕ 8 arctan 1 2 sin 2 8 displaystyle phi theta arctan left tfrac 1 2 sin 2 theta right Contents 1 Tangents via infinitesimals 2 Derivative 3 References 4 External linksTangents via infinitesimals editThe tangent lines of the kappa curve can also be determined geometrically using differentials and the elementary rules of infinitesimal arithmetic Suppose x and y are variables while a is taken to be a constant From the definition of the kappa curve x 2 x 2 y 2 a 2 y 2 0 displaystyle x 2 left x 2 y 2 right a 2 y 2 0 nbsp Now an infinitesimal change in our location must also change the value of the left hand side so d x 2 x 2 y 2 a 2 y 2 0 displaystyle d left x 2 left x 2 y 2 right a 2 y 2 right 0 nbsp Distributing the differential and applying appropriate rules d x 2 x 2 y 2 d a 2 y 2 0 2 x d x x 2 y 2 x 2 2 x d x 2 y d y a 2 2 y d y 0 4 x 3 2 x y 2 d x 2 y x 2 2 a 2 y d y 0 x 2 x 2 y 2 d x y x 2 a 2 d y 0 x 2 x 2 y 2 y a 2 x 2 d y d x displaystyle begin aligned d left x 2 left x 2 y 2 right right d left a 2 y 2 right amp 0 6px 2x dx left x 2 y 2 right x 2 2x dx 2y dy a 2 2y dy amp 0 6px left 4x 3 2xy 2 right dx left 2yx 2 2a 2 y right dy amp 0 6px x left 2x 2 y 2 right dx y left x 2 a 2 right dy amp 0 6px frac x left 2x 2 y 2 right y left a 2 x 2 right amp frac dy dx end aligned nbsp Derivative editIf we use the modern concept of a functional relationship y x and apply implicit differentiation the slope of a tangent line to the kappa curve at a point x y is 2 x x 2 y 2 x 2 2 x 2 y d y d x 2 a 2 y d y d x 2 x 3 2 x y 2 2 x 3 2 a 2 y d y d x 2 x 2 y d y d x 4 x 3 2 x y 2 2 a 2 y 2 x 2 y d y d x 2 x 3 x y 2 a 2 y x 2 y d y d x displaystyle begin aligned 2x left x 2 y 2 right x 2 left 2x 2y frac dy dx right amp 2a 2 y frac dy dx 6px 2x 3 2xy 2 2x 3 amp 2a 2 y frac dy dx 2x 2 y frac dy dx 6px 4x 3 2xy 2 amp left 2a 2 y 2x 2 y right frac dy dx 6px frac 2x 3 xy 2 a 2 y x 2 y amp frac dy dx end aligned nbsp References editLawrence J Dennis 1972 A Catalog of Special Plane Curves New York Dover pp 139 141 ISBN 0 486 60288 5 External links editWeisstein Eric W Kappa curve MathWorld A Java applet for playing with the curve O Connor John J Robertson Edmund F Kappa Curve MacTutor History of Mathematics Archive University of St Andrews Retrieved from https en wikipedia org w index php title Kappa curve amp oldid 1212969865, wikipedia, wiki, book, books, library,
