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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] ${\left({x}^{2}+{y}^{2}\right)}^{2}-4xn\left({x}^{2}-{y}^{2}\right)=0$

where $n$ is a natural integer, that is $n\in \mathrm{ℕ}$ .

Polar coordinates

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

[2] $\rho =4n\mathrm{cos}\left(\theta \right)\mathrm{cos}\left(2\theta \right)$

Parametric formulas

The parametric equations of a siluroid are:

[1a]

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

Cartesian coordinates

In Cartesian coordinates, the mother curve is:

[3] ${\left({x}^{2}+{y}^{2}\right)}^{2}-4x\left({x}^{2}-{y}^{2}\right)=0$

Polar coordinates

In Polar coordinates, the mother curve is:

[4] $\rho =4\mathrm{cos}\left(\theta \right)\mathrm{cos}\left(2\theta \right)$

Solutions

The solutions for the variable y are:

[5] $y=±\sqrt{-{x}^{2}±2\sqrt{{x}^{2}\left(2x+1\right)}-2x}$

Parametric formulas

The parametric equations of the mother curve are:

[3a]

My Profiles

The Siluroid

The Siluroid Curve and Me!