kitchen table math, the sequel: The Conic Sections

Tuesday, October 18, 2011

The Conic Sections

Ah, the conic sections. Like many of the topics in the Pre-Calculus curriculum, this topic can take as much or as little time as you like. The basic ideas are a focus on the four conic sections Parabola, Circle, Ellipse, Hyperbola. There are many different ways to consider the shape and defining components of these mathematical objects.

Conics as a Slice of a Cone

The Greeks originally conceived of these as the shapes generated by slicing through a right circular cone. In A History of Greek Mathematics Vol. II (1921), Sir Thomas Heath says:
The question arises, how did Menaechmus come to think of obtaining curves by cutting a cone? On this we have no information whatever. (pg. 110)

Lost in the mists of time!

More after the jump...

Conics on the Cartesian Plane

In terms of Cartesian analytic geometry, each conic has its respective equation and completing the square is an essential element to this conception of the conics. Each of these general equations requires algebraic perfect squares to analyze the position and shape of the curve. In the case of the parabola, (h,k) represents the vertex, for the circle, ellipse and hyperbola, (h,k) is the center.

Parabola: 4p(y-k)=(x-h)^2
Circle: (x-h)^2 + (y-k)^2 = r^2
Ellipse: (x-h)^2/a^2 + (y-k)^2/b^2 = 1
Hyperbola: (x-h)^2/a^2 - (y-k)^2/b^2 = 1

As I sometimes mention to my students, these "standard" equations sometimes come in different forms in different textbooks. Some texts isolate the y variable in the Parabola equation to arrive at


In many situations it's useful to orient the parabola with a vertex at the origin as:


I typically use this last form for application problems, but use the more general form in studying the general parabola. It's actually interesting that you can square out the general equation of the parabola and collect like terms to show that -b/2a in the quadratic y=ax^2+bx+c does return a value of "h" as the x coordinate of the vertex.

Conics as a Locus of Points

The conics are also defined as loci - that is, as a locus (or collection) of points that have a particular property:

Parabola: a locus of points equidistant from a given point (focus) and a given line (directrix).
Circle: a locus of points equidistant from a given point (center of the circle).
Ellipse: a locus of points the sum of whose distances from two given points (foci) is constant.
Hyperbola: a locus of points the difference of whose distances from two given points (foci) is constant.

The analytic equations can be derived from the locus definitions, mainly by using the distance formula.


Something I have always found amazing about the conics is their appearance in nature. The parabola is an easy one to show as any object thrown into the air traces out a parabola (gravity's rainbow). This can be proven using a little physics and trigonometry:

In the picture above the straight line from lower left to upper right is the path an object would take without gravity (in a vacuum), and the dangling pieces show the position of the object as gravity acts on it, creating the parabolic path. The length of each string in the picture is the height lost due to gravity.

I've done application problems on parabolic trajectory, but in the Pre-Calculus course I currently teach, we focus on the paraboloid of revolution that is used in satellite dishes, headlights and flashlights. These applications depend on the amazing property that all light and radio waves hitting a parabolic dish are reflected to the focus of the dish. Headlights and flashlights use the opposite property that all light emanating from the focus will be reflected off the paraboloid and shine forward in a concentrated beam. I also sometimes mention the parabolic microphone often used in televising football games

The paraboloid shape is also being used to generate electricity from solar radiation. A tower containing salt is placed at the focus and the sun's heat energy is used to melt the salt and heat it to 500 degrees Celsius. The molten salt then runs through a pipe in the middle of a water tank generating steam to drive a turbine.

The other conic application that I find fascinating is the work of Kepler, Brahe and Copernicus that showed that the planets' paths about the sun are elliptical. The Greeks conceived of the ellipse before 300 BCE and Kepler did his work right around 1600 CE - a span of nearly 2000 years between conception and application.

The hyperbola can be used to describe the path a comet might take when passing through our solar system and is also used in the LORAN navigation system to describe the intersection of radio waves.

I generally have about 7 class periods in which to cover the conics, so I focus on completing the square, the general equations for the circle and parabola and applications with the paraboloid.

Here's an interesting project related to the parabola that was inspired by Paul Nahin's book An Imaginary Tale.

The Textbooks

Sullivan covers the circle in Section 2.4, with the rest of the conics in Chapter 11. He gives the equations in a similar form as I've listed above. Coburn gives the same versions of these equations for the circle, ellipse and hyperbola, but goes through some fairly convoluted explanations of what is happening with the equation for the parabola. He ends up presenting the y=ax^2+bx+c, the x^2=4py, and the (x-h)^2=4p(y-k) all together in the same section with appropriate explanations, but I think that this might end up being something of a jumble for a student (or unprepared teacher).

Sobel and Lerner cover the circle in section 4.1 and then cover the ellipse, hyperbola and parabola in chapter 10 (together with non-linear systems). They give the equations in a similar form as I've listed above.

Paul Foerster (typically) is a little different from everyone else - he covers the conics in Chapter 12 with the most general equation of all -
I've left out this form of the conic equations because I've never had the opportunity to teach it before. The xy term rotates a particular conic away from the vertical (or horizontal) orientation it would otherwise have. He follows this up with equations similar to the ones in Sullivan and Coburn but given in an odd form with a "scaling factor" which I've never seen used before.

Brown and Robbins cover the circle in Section 1-6 and the parabola in the form y=ax^2+bx+c in Section 2-3, with the general conics in Chapter 11. His gives his ellipse and hyperbola equations in a similar form as the other texts (except Foerster) but uses the (y-k)=1/(4p)*(x-h)^2 for the parabola.

Dolciani covers the conics in Chapter 13, first presenting the curves with center (or vertex) at the origin, and then using the more general equations to introduce some of the major ideas of transformations in the plane. She ends with the most general form Ax^2+Bxy+Cy^2+Dx+Ey+F=0.


Anonymous said...

Correction: a paraboloid does not reflect all light to the focus, just light rays parallel to the axis of rotation. That's why you have to point the dish at the satellite.

Rich Beveridge said...

Thanks for the correction!

SteveH said...

How about the parametric forms? It took me a long time when I was young to break the dominant influence of explicit function definitions, and I thought that implicit functions were useless. I would have liked (very early on) a discussion of the differences and uses of the explicit, implicit, and parametric forms of equations.

It's easy to start with a line using the standard, implicit form

aX + bY + c = 0

and the explicit slope-intercept form

Y = mX + b

Then you can ask students what they would do with a vertical line. They could create a special case of X= a constant, but what if the slope is near vertical? At what point do you switch to the X=constant form? This would be a good place to introduce the parametric form

X = x1 + (x2-x1)*t
Y = y1 + (y2-y1)*t

where (0<=t<=1)

The benefit of this definition is that you can define line segments exactly and you never have to worry about dividing by a very small number. You can't do that with the other forms.

Also, for the standard form

aX + bY + c = 0

you can check to see if a point is on the line by plugging in an (X,Y) point and seeing if you get zero (within some meaningful tolerance).

If you normalize the equation so that so that a^2 + b^2 = 0, then a and b are a unit vector normal to the line (direction cosines) and c is the signed perpendicular distance from the origin to the line. Also, if you then plug in an (X,Y) value into the equation, it will give you the signed distance of that point to the line. This also works for the equation of a plane.

For conic sections you can show the parametric form of a circle

X = h + r*cos(theta)
Y = k = r*sin(theta)

(start angle <= theta <= end angle)

For hyperbolas, it would be a great way to introduce the cosh and sinh hyperbolic functions.

For a real world application, most CNC cutting machines cut using only straight lines. If you have a curve, you need to subdivide the curve into tiny segments and supply the machine with the end points of the tiny segments. If you use an explicit function form of an equation, what do you do if the slope of the curve approaches infinity? How do you deal with the steps used to evaluate a series of points on the curve? With a parametric form, this is easy. You just have to pick a parametric 't' step size that is small enough to meet your accuracy tolerance.

I know it's possible to overwhelm students with really neat stuff, but I would have liked a more balanced approach to explict, implicit, and paramtric forms of equations.

rocky said...

If you normalize the equation so that so that a^2 + b^2 = 0

You mean a^2 + b^2 = 1

That's pretty zippy. I hadn't seen that before. I'm still trying to prove it. I got the circle part: you're just using polar coordinates, but sinh and cosh got me. I'd only seen them used for complex numbers.

This is high school precalc, right? ;)

SteveH said...

"You mean a^2 + b^2 = 1"

Oops, yes.

There are all sorts of neat tricks when it comes to geometry.

I don't know if sinh and cosh are pre-calc stuff, but my main wish would be to have more emphasis in high school on the different equation forms and why one form might be better than another. I didn't learn this until much later.

SteveH said...

When you plug in an (X,Y) point into the equation it's really the dot product of the position vector (X,Y) and the unit perpendicular vector (a,b) plus c. That works out to be the distance of (X,Y) to the line - as long as a & b are normalized!

ChemProf said...

Yeah, in a perfect world, it would be great if they would spend some time on different forms and vectors. They aren't such difficult topics, and it would be great if students had some experience with them before calculus and physics.

However, often pre-calculus is just a way of making three years of algebra and trig stretch to four, so I don't have a lot of hope.

Glen said...

Steve, I got the approach you're talking about in physics class. I didn't see it in a math class until linear algebra which, as is usual for LA, was after calculus, not before.