Equation general of conic in polar coordinates

In summary, the conic equation has 2 versions in cartesian coordinates: the general form, which can be expressed as ##Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0##, and the parametric form, which can be written as ##y^2 = 2px + (e^2-1)x^2##. In polar coordinates, only the parametric form is known, as ##r = \frac{p}{1+e\cos(\theta)}##. However, it is possible to express the general form in terms of polar coordinates, but it may be complex and messy. The parametric form is preferred due to its simplicity and the fact that it uses
  • #1
Jhenrique
685
4
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?
 
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  • #2
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.
 
  • #3
mfb said:
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...
 
  • #4
Well, you look for the result of a transformation. Isn't it natural to just do this transformation then?
 
  • #5
Ahhh, does not matter... thanks of anyway
 

1. What is the general equation of a conic in polar coordinates?

The general equation of a conic in polar coordinates is r = A + Bcos(θ) or r = A + Bsin(θ), where A and B are constants and r and θ represent the polar coordinates of a point on the conic.

2. How do I convert the general equation of a conic in polar coordinates to rectangular coordinates?

To convert the general equation of a conic in polar coordinates to rectangular coordinates, use the following equations: x = rcos(θ) and y = rsin(θ). Substitute these values for x and y in the general equation and simplify to get the rectangular equation of the conic.

3. What are the different types of conics that can be represented by the general equation in polar coordinates?

The general equation of a conic in polar coordinates can represent a circle, ellipse, parabola, or hyperbola. The type of conic depends on the values of the constants A and B.

4. How is the center of a conic in polar coordinates determined from the general equation?

The center of a conic in polar coordinates is determined by the values of A and B in the general equation. If A = 0, the center is at the origin (0,0). If A ≠ 0, the center is at (A,0) for a conic with a horizontal symmetry axis, or at (0,A) for a conic with a vertical symmetry axis.

5. Can the general equation of a conic in polar coordinates be graphed?

Yes, the general equation of a conic in polar coordinates can be graphed by plotting points with values of r and θ that satisfy the equation. The resulting graph will be a curve that represents the conic.

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