Conic Sections in Polar Form — Theory
1. Polar conic form
A conic section with one focus at the pole can often be written as
\[
r=\frac{\ell}{1\pm e\cos\theta}
\]
or
\[
r=\frac{\ell}{1\pm e\sin\theta}.
\]
Here, \(e\) is the eccentricity and \(\ell\) is the semi-latus rectum.
2. Classification by eccentricity
Eccentricity determines the type of conic:
\[
0\le e<1
\quad\Longrightarrow\quad
\text{ellipse},
\]
\[
e=1
\quad\Longrightarrow\quad
\text{parabola},
\]
\[
e>1
\quad\Longrightarrow\quad
\text{hyperbola}.
\]
The special case \(e=0\) gives a circle centered at the pole.
3. Focus-directrix definition
A conic can be defined as the set of all points whose distance to a focus divided by their distance
to a directrix is constant:
\[
\frac{\text{distance to focus}}{\text{distance to directrix}}
=
e.
\]
In the polar conic forms used here, the focus is at the pole:
\[
F=(0,0).
\]
4. Meaning of the cosine and sine terms
The denominator determines the axis direction:
5. Example: ellipse
Consider
\[
r=\frac{4}{1-0.5\cos\theta}.
\]
Here,
\[
\ell=4,
\qquad
e=0.5.
\]
Since \(0.5<1\), the conic is an ellipse.
6. Ellipse properties
For an ellipse in polar conic form, the semi-major axis is
\[
a=\frac{\ell}{1-e^2}.
\]
The focal distance is
\[
c=ea.
\]
The semi-minor axis is
\[
b=\frac{\ell}{\sqrt{1-e^2}}.
\]
7. Hyperbola properties
For a hyperbola, \(e>1\). A useful set of quantities is
\[
a=\frac{\ell}{e^2-1},
\]
\[
c=ea,
\]
\[
b=\frac{\ell}{\sqrt{e^2-1}}.
\]
Hyperbolas may have angle values where the polar denominator becomes zero. These values correspond to
asymptotic behavior.
8. Parabola properties
When \(e=1\), the conic is a parabola. The focus is at the pole and the directrix is a line.
The vertex lies halfway between the focus and the directrix along the axis.
\[
e=1
\quad\Longrightarrow\quad
\text{parabola}.
\]
9. Cartesian comparison
To compare polar form with Cartesian form, use
\[
x=r\cos\theta,
\qquad
y=r\sin\theta.
\]
For a form
\[
r=\frac{\ell}{1+A\cos\theta+B\sin\theta},
\]
multiply both sides:
\[
r+A x+B y=\ell.
\]
Then square \(r=\sqrt{x^2+y^2}\):
\[
x^2+y^2=(\ell-Ax-By)^2.
\]
10. Directrix line
The directrix depends on the sign and the trigonometric function in the denominator.
For example,
\[
r=\frac{\ell}{1-e\cos\theta}
\]
has a directrix to the left of the pole:
\[
x=-\frac{\ell}{e}.
\]
Different signs and sine/cosine choices rotate or reflect this directrix.
11. Units on graph axes
If \(\ell\), \(x\), and \(y\) are measured in meters, centimeters, feet, or another unit, the graph axes
should display that unit:
\[
x\;(\mathrm{units}),
\qquad
y\;(\mathrm{units}).
\]
This calculator includes an axis-unit field so physical or geometric units stay visible.
13. Common mistakes
- Confusing \(\ell\) with \(a\): \(\ell\) is the semi-latus rectum, not necessarily the semi-major axis.
- Ignoring eccentricity: \(e\) determines whether the conic is an ellipse, parabola, or hyperbola.
- Forgetting the focus: in this polar setup, the focus is at the pole.
- Missing the directrix: focus-directrix identification is central to polar conics.
- Plotting across singular angles: hyperbolas may have denominator values close to zero.
- Dropping units: graph axes should show the same units as the radius and coordinate values.
