Kappa curve

In geometry, the kappa curve or Gutschoven's curve is a two-dimensional algebraic curve resembling the Greek letter κ (kappa).

Related Topics:
Geometry - Algebraic curve - Greek letter - κ (kappa)

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Using the Cartesian coordinate system it can be expressed as:

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:x^4+x^2y^2=a^2y^2

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or, using parametric equations:

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:

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egin{matrix}

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x&=&acos t,cot t\

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y&=&acos t

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end{matrix}

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In polar coordinates its equation is even simpler:

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:r=acot heta

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It has two vertical asymptotes at x=pm a, they have been denoted as blue dashed lines on the graphic.

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The kappa curve's curvature:

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:kappa( heta)={8left(3-sin^2 heta ight)sin^4 hetaover aleft^{3over2}}

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Tangential angle:

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:phi( heta)=-rctanleft

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The kappa curve was first studied by Gérard van Gutschoven around 1662. Other famous mathematicians who have studied it include Isaac Newton and Johann Bernoulli.

Related Topics:
Gérard van Gutschoven - 1662 - Isaac Newton - Johann Bernoulli

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Its tangents were first calculated by Isaac Barrow in the 17th century.

Related Topics:
Tangent - Isaac Barrow - 17th century

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