Parametric Equations

TABLE OF CONTENTS :

TABLE OF CONTENTS

A parametric equation is a way of mapping multiple variables in one variable, so instead of x and y coordinates, we use t coordinates.

Parametric form

Let's look at an example. is a Cartesian equation with coordinates . We can change this assuming that . We now have the coordinates . This is in parameterized form. Parametric equations are very useful in a variety of situations. For example, they can be used in a variety of physics situations where time is used as a variable, and velocity or distance is taken in terms of it.

There are some key parameterizations we can learn to make our lives easier when it comes to writing equations out. We're going to be looking at circles and ellipses in particular.

Circles

Circles have a Cartesian general equation. This equation is for a circle with centre and radius

A key identity we can use as we are squaring two different important values is .

Let's look at an example of parametrizing a circle.

Find a suitable parameterization for the circle .

SOLUTION:

Using

, we want to find a way to write

. So if we let

, this means that

Same way if we let

, this means that

So our answers are

Ellipses

An ellipse is a bit different to a circle. It has an oval shape like this:

This has a general equation , where a is the distance in the x-direction (the semi-major axis) and b is the distance in the y-direction (the semi-minor axis).

We can also parameterize this using . Let's look at an example.

Find a suitable parameterization for the ellipse .

SOLUTION:

This time we want the denominator to cancel out the constant.

Let and .

This means

Therefore

So far we've been looking at transferring from Cartesian form to parametric form. This means moving from our standard x and y relationship to a relationship with t. Now we're going to look the other way around and move from parametric form back to Cartesian form.

How do we write equations in Cartesian form?

We're able to use a few trigonometric identities and work backwards from equations in parametrized form back to equations in Cartesian form. The trigonometric identities we can use are:

Let's look at a difficult example of this:

A curve C has parametric equations where . Find the Cartesian equation of this curve C.

SOLUTION:

Let's use addition formulas to write in a simpler way.

This

therefore,

Therefore:

So is the Cartesian equation of Curve C.

Here we've looked at how to convert parametric equations to Cartesian form. One of the lovely applications of this is that it can make our lives so much easier when finding unclear intersections. We'll see how this helps us in our next section!

How do we find points of intersection?

Points of intersection can be found from parametric equations. This is the same sort of idea as solving equations by looking at intersections on graphs, but this time we are going to use algebraic methods to help us see this.

This can be done by changing between Cartesian and parametric form and solving regularly as you would with simultaneous equations. Intersections are the solutions when two equations are equal.

The parametric equations and generalise a curve C with . The line meets Curve C at point P. Does point P exist?

SOLUTION: Firstly substitute into the Cartesian equation x and y.

Then we can use the trigonometric identity :

Now let's solve for t:

Let

Now to find x simply which gives us:

Now let's find x and y from our t values:

So our first point of intersection is

To find our next point of intersection substitute in our other t value.

So our second point of intersection is

Points of intersection are just one type of point in coordinate geometry that our new methods can help us find. Another type of point is a turning point. Turning points are normally found by differentiation, but this can prove difficult in curves that don't have an easy Cartesian form. Luckily, we have a method of parametric differentiation which we will look at next.

Parametric differentiation

We can also differentiate when functions are parameterized easily. This is due to the fact that and are written in terms of t meaning:

is just the reciprocal of

This can be useful – if a graphs is not clear in Cartesian form, we can find the derivative in parametric form and use the conversion above to find their derivative in Cartesian form.

This is used over converting to Cartesian form then differentiating as it can save time and allows us to get from one point to another very quickly.

Using this, let's look at a few examples of parametric differentiation:

and is a parameterization for a curve C. Find the value of in terms of x and y.

SOLUTION:

other

This means that because

This means that