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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:

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

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:

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

### 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:

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

### Polar coordinates

In Polar coordinates, the mother curve is:

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

### Solutions

The solutions for the variable y are:

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

### Parametric formulas

The parametric equations of the mother curve are:

[3a]

### My Profiles

#### The Siluroid

The Siluroid Curve and Me!