Polar equations of conics
Parabolas, ellipses, and hyperbolas have different definitions in rectangular coordinates. In polar coordinates, one equation covers all three. This is the form astronomers use, because a planet or comet moves on a conic with the Sun at a focus.
One definition for all three
Take a fixed point ${F}$ (the focus), a fixed line ${l}$ (the directrix), and a positive number ${e}$ (the eccentricity). The set of points ${P}$ for which
$\dfrac{\text{distance from }P\text{ to }F}{\text{distance from }P\text{ to }l}=e$
is a conic. It is
- an ellipse if ${e<1}$,
- a parabola if ${e=1}$ (this is the definition from Parabolas),
- a hyperbola if ${e>1}$.
For ellipses and hyperbolas, this ${e}$ is the same eccentricity ${\dfrac{c}{a}}$ as before.
The polar equation
Put the focus at the pole, and the directrix at ${x=d}$ (to the right, ${d>0}$). A point ${P(r,\theta)}$ is at distance ${r}$ from the focus, and at distance ${d-r\cos\theta}$ from the directrix. The definition says ${r=e(d-r\cos\theta)}$. Solving for ${r}$:
${r=\dfrac{ed}{1+e\cos\theta}}$
Other positions of the directrix change the denominator:
- ${1-e\cos\theta}$: directrix ${x=-d}$ (to the left)
- ${1+e\sin\theta}$: directrix ${y=d}$ (above)
- ${1-e\sin\theta}$: directrix ${y=-d}$ (below)
Here is the same focus and directrix with three different eccentricities:
Watch out: To read ${e}$, the denominator must start with ${1}$. If it starts with another number, divide the top and bottom by that number first.
Example 1: Identify the conic ${r=\dfrac{6}{1+\cos\theta}}$, and find its vertex.
Solution:
The denominator starts with ${1}$, and ${e=1}$. So it is a parabola, with ${ed=6}$, so ${d=6}$: the directrix is ${x=6}$.
The vertex is halfway between the focus (the pole) and the directrix. At ${\theta=0}$: ${r=\dfrac{6}{2}=3}$. The vertex is ${(3,0)}$, and the parabola opens to the left.
Example 2: Identify the conic ${r=\dfrac{10}{3-2\cos\theta}}$, and find its vertices.
Solution:
Divide the top and bottom by ${3}$:
${r=\dfrac{10/3}{1-\dfrac{2}{3}\cos\theta}}$
So ${e=\dfrac{2}{3}<1}$: an ellipse. From ${ed=\dfrac{10}{3}}$, ${d=5}$, and the minus sign puts the directrix at ${x=-5}$.
The vertices are on the polar axis. At ${\theta=0}$: ${r=\dfrac{10}{3-2}=10}$, the point ${(10,0)}$. At ${\theta=\pi}$: ${r=\dfrac{10}{3+2}=2}$, the point ${(-2,0)}$.
Check: The major axis runs from ${-2}$ to ${10}$, so ${a=6}$ and the center is ${(4,0)}$. The focus at the pole is ${c=4}$ from the center, and
${\dfrac{c}{a}=\dfrac{4}{6}=\dfrac{2}{3}=e}$
Example 3: Identify the conic ${r=\dfrac{12}{2+4\sin\theta}}$, and find its vertices.
Solution:
Divide by ${2}$: ${r=\dfrac{6}{1+2\sin\theta}}$. So ${e=2>1}$: a hyperbola. From ${ed=6}$, ${d=3}$; the directrix is ${y=3}$.
The vertices are on the vertical line through the focus. At ${\theta=\dfrac{\pi}{2}}$: ${r=\dfrac{6}{3}=2}$, the point ${(0,2)}$. At ${\theta=\dfrac{3\pi}{2}}$: ${r=\dfrac{6}{1-2}=-6}$. A negative ${r}$ in the direction ${\dfrac{3\pi}{2}}$ (down) means ${6}$ units up: the point ${(0,6)}$.
Orbits
Johannes Kepler discovered around 1609 that each planet moves on an ellipse with the Sun at one focus. Isaac Newton later showed that every object moving under the Sun's gravity alone follows a conic with the Sun at a focus: an ellipse for planets and returning comets, and a parabola or hyperbola for objects that pass by once and never return.
For a planet, the closest point to the Sun is the perihelion and the farthest is the aphelion. They are the two vertices, at ${\theta=0}$ and ${\theta=\pi}$ in the polar equation.
Summary
- With focus at the pole: ${r=\dfrac{ed}{1\pm e\cos\theta}}$ or ${r=\dfrac{ed}{1\pm e\sin\theta}}$.
- ${e<1}$: ellipse. ${e=1}$: parabola. ${e>1}$: hyperbola.
- Make the denominator start with ${1}$ before reading ${e}$ and ${d}$.
- The vertices come from ${\theta=0}$ and ${\pi}$ (cosine), or ${\dfrac{\pi}{2}}$ and ${\dfrac{3\pi}{2}}$ (sine).