When we first learn the equation of an ellipse, we usually encounter its Cartesian equation:
This equation describes an ellipse centered at the origin.
Here, a represents the horizontal semi-axis and b represents the vertical semi-axis.
Later, when studying parametric equations, we encounter the parametrization:
At first, these equations may seem unrelated to the standard equation of an ellipse.
Why should x involve cosine? Why should y involve sine? And why do we multiply them by a and b?
There is a very natural connection between the familiar parametrization of a circle and the parametrization of an ellipse.
Start with a circle
It is helpful to begin with the unit circle.
The equation of the unit circle is:
A point moving around this circle can be described using an angle t.
Its coordinates are:
This is the familiar parametric representation of a circle.
Why do these formulas work?
Because for every value of t:
Using the fundamental trigonometric identity:
we obtain:
So the point (cos t, sin t) always lies on the unit circle.
The idea of stretching the circle
Now imagine taking the unit circle and stretching it horizontally.
Suppose every horizontal coordinate is multiplied by a.
The equation for the x-coordinate becomes:
Next, suppose we stretch the circle vertically by a factor of b.
The y-coordinate becomes:
Therefore, after stretching the unit circle horizontally by a and vertically by b, we obtain:
This is the parametric equation of the ellipse.
So one way to remember the idea is:
Start with the unit circle and stretch it horizontally and vertically.
The unit circle has:
The ellipse has:
The factors a and b control the amount of stretching.
Verify that it really gives an ellipse
We can prove this algebraically.
Start with the parametric equations:
Square both equations:
Now divide the first equation by a²:
And divide the second equation by b²:
Add the two equations:
Using:
we get:
This is exactly the standard equation of the ellipse.
Therefore, the parametric equations do indeed describe the ellipse.
Why do we use a parameter?
The parameter t tells us where the point is on the ellipse.
As t changes, x and y change.
Consequently, the point (x,y) moves around the ellipse.
This is an important difference between a Cartesian equation and a parametric equation.
The Cartesian equation:
describes the relationship between x and y.
The parametric equations:
describe how the coordinates are generated as the parameter t changes.
Look at the familiar angles
Just as with the circle, certain values of t are particularly easy to understand.
Consider t = 0.
Since cos 0 = 1 and sin 0 = 0:
Therefore:
This is the rightmost point of the ellipse.
Now consider t = π/2.
Since cos(π/2) = 0 and sin(π/2) = 1:
Therefore:
This is the topmost point.
At t = π:
Therefore:
This is the leftmost point.
At t = 3π/2:
Therefore:
This is the bottommost point.
Finally, at t = 2π:
So the point returns to:
The point has completed one complete trip around the ellipse.
A table of the important points
The movement can be summarized as follows:
| t | x | y | Point |
|---|---|---|---|
| 0 | a | 0 | (a, 0) |
| π/2 | 0 | b | (0, b) |
| π | -a | 0 | (-a, 0) |
| 3π/2 | 0 | -b | (0, -b) |
| 2π | a | 0 | (a, 0) |
These are useful checkpoints for understanding the motion.
However, just as with the circle, the ellipse contains infinitely many other points.
What happens between the checkpoints?
This is where the parameter becomes particularly useful.
Suppose t starts at 0 and gradually increases.
At t = 0:
As t increases, the point begins moving upward and to the left.
For example, at t = π/4:
and:
Therefore:
This point lies between (a,0) and (0,b).
As t continues to increase, the point continuously travels around the ellipse.
There is no jump from one checkpoint to another.
The parametric equations generate all the intermediate points as well.
Why are the horizontal and vertical coordinates different?
This is an important feature of the ellipse.
For a circle of radius r, both coordinates are multiplied by the same radius:
But an ellipse generally has different horizontal and vertical dimensions.
Its horizontal radius is a.
Its vertical radius is b.
Therefore, the horizontal coordinate needs the factor a:
while the vertical coordinate needs the factor b:
If a and b are different, the resulting curve is stretched differently in the two directions.
That is what produces the ellipse.
What if a = b?
This gives us an interesting special case.
Suppose:
Then the parametric equations become:
These are exactly the parametric equations of a circle of radius r.
Therefore, a circle can be considered a special case of an ellipse in which the horizontal and vertical semi-axes are equal.
If a ≠ b, we generally get an ellipse.
If a = b, we get a circle.
The deeper connection with the unit circle
There is an even more intuitive way to view the parametrization.
Start with a point on the unit circle:
Now multiply its x-coordinate by a and its y-coordinate by b.
The point becomes:
This is simply a coordinate transformation.
The x-coordinate has been stretched by a.
The y-coordinate has been stretched by b.
So:
Unit circle:
Ellipse:
This is perhaps the simplest geometric way to understand why the ellipse has this parametrization.
Connecting this to arc length
The parametrization is not only useful for describing the ellipse.
It also allows us to calculate quantities such as its arc length.
For a general parametric curve:
the differential arc length is:
For the ellipse:
we differentiate:
and:
Therefore:
which simplifies to:
Since t goes from 0 to 2π for one complete revolution, the circumference is:
Thus, the parametrization of the ellipse becomes the starting point for its arc-length calculation.
A useful way to remember the whole idea
We can summarize the development in a few steps.
Step 1: Start with the unit circle
Step 2: Stretch horizontally
Multiply the x-coordinate by a:
Step 3: Stretch vertically
Multiply the y-coordinate by b:
Step 4: Obtain the ellipse
The resulting point is:
Step 5: Verify it
Substitution gives:
which is the equation of the ellipse.
The key conceptual distinction
It is tempting to think:
“We use the points at 0°, 90°, 180°, and 270° to create the parametric equation.”
That is not quite correct.
Those familiar points are checkpoints that help us understand the curve.
They show us what happens at particularly simple values of the parameter.
But an ellipse contains infinitely many other points.
The trigonometric functions give us the general rule that works for every value of t:
The parameter can take infinitely many values, and each value produces a corresponding point on the ellipse.
Therefore:
The special angles help us visualize the pattern.
The trigonometric functions express the general pattern.
The parameter allows us to trace the entire ellipse.
From a circle to an ellipse
The entire idea can ultimately be understood through one simple transformation.
Start with:
which traces the unit circle.
Then stretch the x-coordinate by a and the y-coordinate by b:
The result is an ellipse.
So the parametrization:
is not an arbitrary formula that we simply memorize.
It comes naturally from the parametrization of the circle and the geometric idea of stretching that circle differently in the horizontal and vertical directions.
Final takeaway
The Cartesian equation:
tells us which points belong to the ellipse.
The parametric equations:
tell us how to generate and trace those points as the parameter t changes.
The most important connection to remember is:
Unit circle:
Stretch horizontally by a and vertically by b:
Result: an ellipse.
This is why x = a cos t and y = b sin t are such a natural and useful parametrization of an ellipse.
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