Knot Curve
KnotCurve
The knot curve is a quartic curve with implicit Cartesian equation
| (x^2-1)^2=y^2(3+2y). |
(1)
|
The x- and y-intercepts are (0,-1), (0,1/2), and (+/-1,0). It has horizontal tangents at (0,1/2) and (+/-1,-3/2), and vertical tangents at (+/-sqrt(2),-1).
Its curvature is implicitly given by
| kappa(x,y)=-(6(2x^2-4x^4+2x^6+4x^2y-8x^4y+4x^6y+3y^2-9x^2y^2+6y^3-18x^2y^3+3y^4-9x^2y^4))/((4x^2-8x^4+4x^6+9y^2+18y^3+9y^4)^(3/2)). |
(2)
|
KnotCurveArea
The areas A_1 enclosed by a side region and by the center region A_2 are given by
A_1 = (12)/7sqrt(3)(sqrt(2)-1)
(3)
A_2 = (12)/7sqrt(3)(2-sqrt(2)),
(4)
giving the entire area as
A = 2A_1+A_2
(5)
= (12)/7sqrt(6).
(6)
See also
BowExplore with Wolfram|Alpha
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References
Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 72, 1989.Referenced on Wolfram|Alpha
Knot CurveCite this as:
Weisstein, Eric W. "Knot Curve." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/KnotCurve.html