Conic Section

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Conic Section
Field: Geometry
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Conic Section

A conic section is a curve created from the intersection of a plane with a cone.

Basic Description

When a plane intersects a cone, the main characteristic that affects the resulting curve or conic section is the angle of the plane in relation to the cone.

There are four different types of conic sections

• Circles
• Ellipses
• Parabolas
• Hyperbolas

By changing the angle of the plane in relation to the cone, it can also produce a point, a line, or two intersecting lines. These are called degenerate cases and are not considered conic sections.

Circles

If the plane intersects the cone perpendicular to the axis of the cone, then the curve produced will be a circle.

Ellipses

If the plane intersects the cone at an angle greater than perpendicular to the axis but less than that of the line with the slope of the cone, the curve produced will be an ellipse.

Parabola

If the plane intersects the cone at the same angle as that of the line of the slope of the cone, then the curve produced will be a parabola.

Hyperbola

If the plane intersects the cone at an angle greater than that of the line of the slope of the cone, then the curve produced will be a hyperbola.

Demonstration

Click to show the interactive applet: [show more][hide]

ERROR: Unable to find Java Applet file: ConicSection.class.

A More Mathematical Explanation

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Formal Definition

There is a more formal definition for conic sections.

A conic section ca [...]

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Formal Definition

There is a more formal definition for conic sections. A conic section can be defined as the locus of points P in a plane such that the ratio of the distance between P and a fixed point, called the focus to the distance between P and a fixed line (which does not contain the focus), called the directrix, is a constant value, called the eccentricity. Eccentricity can also be thought of as how much the conic section deviates from being circular.

The focus is usually denoted F, the directrix, d, and the eccentricity, e.

Note that in the simple case of a parabola, the points are defined as all points equidistant between the focus and the directrix. Since these points are equidistant, e=1. In the image below, the focus is a purple dot, the parabola is blue, and the directrix is red. The other colored lines show the distance between the focus and points on the parabola is the same as the distance between those same points and the directrix.

For an ellipse, the points on the ellipse are always closer to the focus than they are to the directrix. If we call the distance between the directrix and a point on the hyperbola L, then we have that the distance between the focus and the directrix is L times the eccentricity, e. Since the eccentricity is less than one, Le is always less than L.

In this image, you can see that the green lines, that go from the ellipse to the foci, are each shorter than corresponding the pink lines, that go from the ellipse to the directrices.

Notice that two foci and two directrices are drawn in here. The points on the ellipse to the left of the y axis are defined by the directrix and focus on the left side of the axis. Similarly, the points on the ellipse to the right of the axis are defined by the directrix and focus on the right side of the axis.

For a hyperbola, the opposite is true: all points on the hyperbola are closer to the directrix than they are to the focus. This is because e is greater than one, so Le is always greater than L for a point on the hyperbola. Like the ellipse, there are two sets of foci and directrices, and each defines one side of the hyperbola.

In this image, the green lines from the hyperbola to the foci are each longer than corresponding the pink lines from the hyperbola to the directrices.

Now, the remaining conic section is the circle. Since eccentricity can be thought of how not circular something is, one can guess that the eccentricity is zero. But that begs the question, where is the directrix if the ratio of these two numbers must be zero?

The center of a circle is its focus, so we know that the distance from any point on the circle to the focus is r, the radius. In order to have the ratio of r to something be zero, we must have that something be infinity. Therefore, we say that the directrix of a circle is a line at infinity.

Algebraic Perspective

Algebraically, a conic section is always represented by an equation of second degree. The general equation for a conic section is the following:
$Ax^2+Bxy+Cy^2+Dx+Ey+F=0$
with A, B, and C not all zero.

The type of conic section can be found using the value of $B^2-4AC$

• If $B^2-4AC<0$ then the section represented by the equation is an ellipse
• If $A=C$ and $B=0$ then the section represented by the equation is a circle
• If $B^2-4AC=0$ then the section represented by the equation is a parabola
• If $B^2-4AC>0$ then the section represented by the equation is a hyperbola

Properties of Conic Sections

Below is a table that summarizes basic information about each of the four conic sections each with a center (0,0):

The equations of conic sections described above can rearranged algebraically into the forms in the chart below.

	Circle	Ellipse	Parabola	Hyperbola
Equation	$x^2+y^2=r^2$	$\frac{y^2}{a^2}+\frac{x^2}{b^2}=1$	$4py=x^2$ or $y=ax^2$	$\frac{y^2}{a^2}-\frac{x^2}{b^2}=1$
Equations of Asymptotes	none	none	none	$x=\pm(\frac{b}{a})y$
Variables	r = circle radius	a = major radius b = minor radius c = distance center to focus	p = distance from vertex to focus (or directrix)	a = $\tfrac{1}{2}$ length major axis b = $\tfrac{1}{2}$ length minor axis c = distance center to focus
Focus	(0,0)	$(\pm \sqrt{a^2-b^2}, 0)$	$(0,\frac{1}{4a})=(0,p)$	$(\pm \sqrt{a^2+b^2}, 0)$
Directrix	The directrix of a circle is any line in the same plane that is infinitely far away.	$x=\pm \frac{\sqrt{a^2-b^2}}{1-b^2/a^2}$	$y=-\frac{1}{4a}=-p$	$x=\pm \frac{\sqrt{a^2+b^2}}{1+b^2/a^2}$
Relation to Focus	$p=0$	$a^2-b^2=c^2$	$p=p$	$a^2+b^2=c^2$
Eccentricity	$e=0$	$0<e=\sqrt{1-b^2/a^2}<1$	$e=1$	$e=\sqrt{1+b^2/a^2}>1$

Numerical Examples

For a parabola, if we let our equation be $4y=x^2$, we end up with the parabola to the right.

We can see that $p=1$ from our equation in the chart.

Also from those, we can see that the focus is at $(0,p)=(0,1)$ (shown in purple) and that the equation of the directrix is $y=-p=-1$ (shown in red).

For an ellipse, let us take our equation to be $\left(\frac{x}{5}\right)^2+\left(\frac{y}{3}\right)^2=1$. This gives us that $a=5$ and $b=3$. This ellipse is shown to the left in blue.

From the equations in the chart, we get that the focal points are at

$(\pm \sqrt{a^2-b^2},0)=(\pm \sqrt{5^2-3^2},0)=(\pm 4,0)$.

These are shown in purple.

The directrices are at

$x=\pm \frac{\sqrt{a^2-b^2}}{1-b^2/a^2}=\pm \frac{\sqrt{5^2-3^2}}{1-3^2/5^2}=\pm \frac{4}{1-9/25}=\pm \frac{4}{16/25}=\pm \frac{25}{4}=\pm 6.25$

These are shown in red.

For a hyperbola, take the equation $\left(\frac{x}{4} \right)^2-\left(\frac{y}{3}\right)^2=1$. This hyperbola is shown in blue on the right.

From the equation for the hyperbola, we see that $a=4$ and $b=3$. Therefore, from the equations, we get that the foci are at the points

$(\pm \sqrt{a^2+b^2},0)=(\pm \sqrt{4^2+3^2},0)=(\pm 5,0)$. These are the purple dots, just as with the ellipse.

Also from the equations, the directrices are at

$x=\pm \frac{\sqrt{a^2+b^2}}{1+b^2+a^2}=\pm \frac{\sqrt{4^2+3^3}}{1+3^2/4^2}=\pm \frac{5}{1+9/16}=\pm \frac{5}{25/16}=\pm \frac{16}{5}=\pm 3.2$.

These are shown in red.

For a circle, there aren't any properties to calculate. We can see in the image on the left that the radius is two and the focus is (0,0). Therefore, the equation of this circle is $x^2+y^2=2^2$

Applications to orbits

The orbits of planets or object orbiting the sun are approximately conic sections with a focus at the sun. Bounded orbits are where the object keeps going around the sun, and they approximately take the form of circles and ellipses. Unbounded orbits are where the object escapes the sun's gravitational field and never comes back. These are parabolas and hyperbolas.

As the famous astronomer and physicist Kepler determined, orbits must take these shakes as a consequence of the fact that the force of gravity is proportional to one over the distance squared.

He proved that the equations of all objects orbiting another, much more massive one, obey the following equation:

$r(\theta)=\frac{c}{1+e \cos \theta}$.

This equation gives the radius of the orbit ($r$) as a function of the angle in radians as measured from the positive (right) side of the x axis ($\theta$). The $e$ in the equation is the same eccentricity as in all of our other equations.

Therefore, when $e=0$ the Kepler orbit equation simply becomes $r(\theta)=\frac{c}{1}=c$. Since this means that the radius is always a constant c, we can see that this type of orbit is circle with radius c.

When $0<e<1$, we have a maximum radius and a minimum radius. These happen when $\cos\theta = -1$ and $\cos \theta = 1$, respectively.

[click to show example]

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Take $e$ to be 0.3 and $c$ to be 2. Then, plotting $r$ as a function of $\theta$ we get the ellipse below.

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The maximum value of $r$ is $r=\frac{2}{1-0.3}=2.86$.

The minimum value of $r$ is $r =\frac{2}{1+0.3}=1.54$.

Each of those can be seen in the image above.

Now, what happens if $e=1$?

Then, when $\cos \theta = - 1$ (this happens when $\theta=\pm \pi$, the denominator of the orbit equation is zero. This means that $r$ approaches infinity. This is precisely the condition when an orbiting object escapes the gravitational field of the larger object, and we get a parabola. $r$ is at a minimum at the vertex of the parabola, and the value is $r_{min}=c/2$.

[click to show example]

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If we let $c=1$, then we have a parabola opening to the left with vertex (0.5,0).

Similarly, when $e>1$, there will be some $\theta_{max}$ and a $\theta_{min}$ that can be found by finding when the denominator of the equation is zero, ie when $-1=e \cos \theta$. Here, we get a hyperbola.

[click to show example]

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If we let $e=2$ and $c=1$ we find that the maximum and minimum values of $\theta$ occur when $-1=2 \cos \theta$, or when $\theta=\cos^{-1}(-1/2)=\pm 2.09$ radians.

We can also find $r_{min}$. Since $e>1$, the smallest value of the denominator is when $\cos \theta =-1$, or at $\theta=0$. So $r_{min}=\frac{1}{1-2(-1)}=1/3$.

The Earth's orbit is quite circular--it's eccentricity is only 0.017. Mercury's orbit is much more elliptical with an eccentricity of more than 0.2. Halley's comet has a very eccentric orbit: with an eccentricity of 0.967, it is very close to having the eccentricity of a parabola and escaping the solar system.

If you want to play around with the eccentricities of orbits, this website as well as this one have cool interactive tools.

Teaching Materials

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References

http://math2.org/math/algebra/conics.htm

http://en.wikipedia.org/wiki/Conic_section

http://mathworld.wolfram.com/ConicSection.html

http://www.2dcurves.com/conicsection/conicsection.html

http://www.windows.ucar.edu/tour/link=/our_solar_system/planets_table.html

http://xahlee.org/SpecialPlaneCurves_dir/ConicSections_dir/conicSections.html

Future Directions for this Page

• A proof or derivation of the general equation for conic sections
• Interactive applet showing the formation of a conic section
  

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