# Department of Mathematics University of Leicester

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Department of Mathematics University of Leicester
Parametric Department of Mathematics University of Leicester

What is it? A parametric equation is a method of defining a relation using parameters. For example, using the equation: We can use a free parameter, t, setting: and

What is it? We can see that this still satisfies the equation, while defining a relationship between x and y using the free parameter, t.

Why do we use parametric equations
Parameterisations can be used to integrate and differentiate equations term wise. You can describe the motion of a particle using a parameterisation: r being placement.

Why do we use parametric equations
Now we can use this to differentiate each term to find v, the velocity:

Why do we use parametric equations
Parameters can also be used to make differential equations simpler to differentiate. In the case of implicit differentials, we can change a function of x and y into an equation of just t.

Why do we use parametric equations
Some equations are far easier to describe in parametric form. Example: a circle around the origin Cartesian form: Parametric form:

How to get Cartesian from parametric
Getting the Cartesian equation of a parametric equation is done more by inspection that by a formula. There are a few useful methods that can be used, which are explored in the examples.

How to get Cartesian from parametric
Example 1: Let: So that: and

How to get Cartesian from parametric
Next set t in terms of y: Now we can substitute t in to the equation of x to eliminate t.

How to get Cartesian from parametric
Substituting in t: Which expands to:

How to get Cartesian from parametric
Example 2: Let: So that: and

How to get Cartesian from parametric
To change this we can see that: And

How to get Cartesian from parametric
And as we know that We can see that:

How to get Cartesian from parametric
Which equals: This is the Cartesian equation for an ellipse.

Example Example 3: let: Be the Cartesian equation of a circle at the point (a,b). Change this into parametric form.

Example If we set: And: Then we can solve this using the fact that:

Example From this we can see that: So: Therefore:

Example Similarly: So: Therefore:

Example Compiling this, we can see that:
Which is the parametric equation for a circle at the point (a,b).

Polar co-ordinates Parametric equations can be used to describe curves in polar co-ordinate form: For example:

Polar co-ordinates Here we can see, that if we set t as the angle, then we can describe x and y in terms of t: Using trigonometry: and

Polar co-ordinates These can be used to change Cartesian equations to parametric equations:

Polar co-ordinates: example
Let: Be the equation for a circle. If we set:

Polar co-ordinates: example
We can see that if we substitute these in, then the equation still holds: Therefore we can use: As a parameterisation for a circle.

Finding the gradient of a parametric curve
To find dy/dx we need to use the chain rule:

How to get Cartesian from parametric: example
Let: and Then:

How to get Cartesian from parametric: example
Then, using the chain rule:

Extended parametric example
Let: Be the Cartesian equation.

Extended parametric example
Then to change this into parametric form, we need to find values of x and y that satisfy the equation. If we set: And:

Extended parametric example
Then we have: Which expands to:

Extended parametric example
We know that: Therefore we can see that our values of x and y satisfy the equation. Therefore:

Extended parametric example
Now, as this is the placement of the particle, we can find the velocity of the particle by differentiating each term:

Extended parametric example
Next, we can find the gradient of the curve. Using the formula:

Extended parametric example
Using this: And:

Extended parametric example