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Parametric

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?.

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Parametric

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  1. 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 • Example: • Let: and Then: and

  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)

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