1. Department of Mathematics University of Leicester
Parametric
Department of Mathematics
University of Leicester

2. 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

3. 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.

4. 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.

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

6. 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.

7. 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:

8. 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.

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

10. 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.

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

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

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

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

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

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

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

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

19. Example
Similarly:
So:
Therefore:

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

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

22. 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

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

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

25. 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.

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

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

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

29. Extended parametric example
Let:
Be the Cartesian equation.

30. 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:

31. Extended parametric example
Then we have:
Which expands to:

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

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

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

35. Extended parametric example
Using this:
And:

36. Extended parametric example
Therefore the gradient is:

37. Conclusion
Parametric equations are about changing equations to just 1 parameter, t.
Parametric is used to define equations term wise.
We can use the chain rule to find the gradient of a parametric equation.

38. Conclusion Standard parametric manipulation of polar co- ordinates is:
x=rcos(t)
Y=rsin(t)