Associated Topics || Dr. Math Home || Search Dr. Math

### Graphing an Ellipse

```
Date: 11/20/98 at 20:23:50
From: Phil
Subject: Coordinate geometry

A family friend was helping me with my math homework, and mentioned
that once, while she was working with coordinate geometry in her senior
year, she graphed a "really nice" ellipse.  She says she forgot how,
and my math teacher says I won't get to that until a lot later. How do
you graph an ellipse? What is the equation? Is it really coordinate
geometry, or does it just involve coordinates?
```

```
Date: 01/08/99 at 13:28:01
From: Doctor Ujjwal
Subject: Re: Coordinate geometry

Dear Phil,

I am glad that you are attracted by the beauty of geometric shapes.
Indeed there are many ways of constructing nice ellipses. The equipment
used may vary from two pins and a string to computers. But since you
are trying to graph one, let's see a graphical method. Remember that
this is just one of the many, many constructions.

A very straightforward method is to use one of the many forms of the
equation of an ellipse. For example:

x^2   y^2
--- + --- = 1
a^2   b^2

where a and b are constants. The greater of the two is called the
semi-major axis and the other is the semi-minor axis. Values of (x, y)
can be calculated from such an equation, plotted on graph paper, and
then connected to get an ellipse, but this is a tedious method so let's
leave it to computers.

There is a fun way of plotting an ellipse. It also brings out the
essential nature of the ellipse as a circle 'stretched' in one
direction.

Draw the horizontal (x) axis and vertical (y) axes on graph paper. The
point (O) where the two axes meet is called the origin. With a compass
draw a circle centered at the origin. Next we are going to stretch this
circle in the x direction. Read the coordinates (x, y) of any point (C)
on the circle. Plot a new point (E) with coordinates (2*x, y). Point
(E) is on an ellipse we would get by stretching the circle to double
its size in the x direction.

^ Y-axis
|           C(x,y)      E(2*x,y)
y|<--- x --->*<--- x --->*
|
|
-|------------------------------> X-axis

Repeat it for more points on the circle and you will see an ellipse
taking shape.

If you get tired of calculations, you can use the divider in your
compass box to mark the x coordinate of (C) and mark it horizontally
from (C) to get point (E). You can also save a lot of time by using the
symmetry of an ellipse about the X and Y axes.

^ Y axis
E1               |                E
*<----- x ------>|<----- x ------>*
|                |                |
y                |                y
|                |                |
------------------|------------------------------> X axis
|                |                |
y                |                y
|                |                |
*<----- x ------>|<----- x ------>*
E3               |                E2

Can you figure out how you can reflect the point (E) about the X and Y
axes to get three more points (E1, E2, E3) on the ellipse? Here too the
divider comes in very handy.

Good luck and happy plotting!

- Doctor Ujjwal Rane, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Conic Sections/Circles
High School Coordinate Plane Geometry
High School Equations, Graphs, Translations
High School Geometry

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search