Equation general of conic in polar coordinates

[Jhenrique]

The conic equation has 2 versions in cartesian coordinates:

The general: $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$
And the parametric: $y^2 = 2px + (e^2-1)x^2$

In polar coordinates, I known just the parametric: $r = \frac{p}{1+e\cos(\theta)}$
But exist a general form too?

 

[mfb]

You can express x and y in your general equation in terms of r and θ to get the general form. I think it will look quite messy. The parametric uses the "right" coordinate system, where the equation has a nice form. Usually there is no point in polar coordinates if the center does not have a special meaning.

 

[Jhenrique]

You can express x and y in your general equation in terms of r and θ to get the general form. I think it will look quite messy. The parametric uses the "right" coordinate system, where the equation has a nice form. Usually there is no point in polar coordinates if the center does not have a special meaning.

But what you are suggesting is a transformation... I'm not looking for this, but yes by a general expression/format that is a conic but is independent of the cartesian format...

 

[mfb]

Well, you look for the result of a transformation. Isn't it natural to just do this transformation then?

 

[Jhenrique]

Ahhh, does not matter... thanks of anyway