Math Is Fun Forum

John

Guest

Ellipse

I'm going to do a course Matrices and Linear Transformations and I am required to know about the Ellipse and Circle Formulae, except I've never even heard of these before. Can someone please link me to a detailed guide explaining what these are, and what they are used for, etc.

Zhylliolom

Real Member

Registered: 2005-09-05

Posts: 412

Re: Ellipse

A circle centered at (h,k) with radius r is given in Cartesian coordinates by the equation

An ellipse centered at (h,k) with semimajor axis a and semiminor axis b is given in Cartesian coordinates by the equation

If you're taking course on matrices and linear transformations, perhaps this page will have better information for you:

[url=]http://en.wikipedia.org/wiki/Matrix_representation_of_conic_sections[/url]

John

Guest

Re: Ellipse

Thanks for the fast reply, but I've never done anything on conics before and don't really understand what you've written. If possible can you please explain or link to some where that explains in detail about conics, I'm only 16 and not doing Matrices and Linear Algebra until next year but my teacher said students are expected to already understand about the Circle and Ellipse Formulae (and presumably conics in general) before we do the unit next year, so I'm trying to find out about these sooner rather than later so I can fully understand what we do next year.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Ellipse

John, do you understand graphing, such as:

y = 2x + 4
y = x²
y = cos(x)
y = √x³
...

Just trying to get  a feel for what you know.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

John

Guest

Re: Ellipse

I could graph all of those except possibly √x³ although I can graph cubic and square root functions.

John

Guest

Re: Ellipse

Please can someone help me!

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Ellipse

Sorry, forgot about this post.

It turns out that if you take the equation:

You get the top half of a circle of radius r, centered around the origin.  Remember, when we do the square root, by notational standards, we only include the positive result.   So to get the bottom half, we use:

And then graphing y_1 and y_2, we get a circle.

But this notation is horrible, we don't want to have two functions to describe a circle, we really just want an equation.  Do you know what the verticle line test is?  Assuming you do, to be a function, it must pass the verticle line test.  It should be obvious that a circle would never pass such a test.  So we should probably abandon trying to write a circle as a function, as it must be impossible.

So we have:

Now that's pretty ugly, so let's try to get ride of the square root:

Now both our x and y are variables, and r is our constant.  So just to make it neater, we want to have our variables on one side, and our constants on another:

That's better.  So this is the equation (not function) of circle centered around the origin with radius r.  That's pretty good, but it isn't enough.  Why must we have it centered around the origin?  I dunno, so lets make it not.

You may not know this, but if you have a function:

y = f(x) (for example, f(x) = 2x or f(x) = cos(x))

Then if we want to shift it up by k, we do:

y = f(x) + k

Notice that if we make k negative, it's the same as shifting it down.  Also, if we want to shift it to the right by h:

y = f(x - h) + k

Again, having a negative h will move it to the left.  So take this and apply it to our circle equation:

Ugly.  Let's take that square root out again.  But remember to subtract k to the right side to make things easier first:

Now we want to get all our variables on the same side again:

And that is the equation of a circle position with a center at (h, k) with radius r.

See if Zhylliolom's post makes sense now.

---

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

John

Guest

Re: Ellipse

Thankyou that makes perfect sense, I know all about the vertical line test and translating graphs so that makes perfect sense now, thanks again!

ben

Member

Registered: 2006-07-12

Posts: 106

Re: Ellipse

Ricky - that was a masterly exposition.

John

Guest

Re: Ellipse

Can someone please show me how you get to the equation of the ellipse, I thought it was just y = a(r^2-(bx-h)^2)^(1/2)+ak rearranged with a and b being the horizontal and vertical dialations but the r^2 disappears altogether and that got me confused.

Zhylliolom

Real Member

Registered: 2005-09-05

Posts: 412

Re: Ellipse

The r[sup]2[/sup] is supposed to disappear. This article outlines the derivation of the equation for an ellipse:

http://en.wikipedia.org/wiki/Derivation … an_ellipse

Sorry for just posting a link again, but this article really covers it completely.

Last edited by Zhylliolom (2006-08-05 18:00:09)

MathsIsFun

Administrator

Registered: 2005-01-21

Posts: 7,711

Re: Ellipse

Ricky - that was a masterly exposition.

I agree, very clearly explained.

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"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman