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Family of curves

Cartesian coordinates

In an x-y Cartesian coordinate system, the most general form of equation of a siluroid is:

[1] x2+y2 2 - 4xn (x2-y2) = 0

where n is a natural integer, that is n .

Polar coordinates

Analogously, the most general form of equation of a siluroid in Polar coordinates is:

[2] ρ = 4 n cosθ cos2θ

Parametric formulas

The parametric equations of a siluroid are:

[1a] x = 2n + t2nty = ±4n2-t22nt where t  -2n, 2n

Mother curve

If n = 1 we obtain the so called mother curve. Note that the mother curve is a particular case of a more general curve called folium. In particular it is a trifolium where a = 12

Cartesian coordinates

In Cartesian coordinates, the mother curve is:

[3] x2+y2 2 - 4x (x2-y2) = 0

Polar coordinates

In Polar coordinates, the mother curve is:

[4] ρ = 4 cosθ cos2θ

Solutions

The solutions for the variable y are:

[5] y= ±-x2 ±2x2 (2x+1) -2x

Parametric formulas

The parametric equations of the mother curve are:

[3a] x = 2 + t2ty = ±4-t22t where t  -2, 2

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