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Paramaterization Of A Curve?

  • 25-04-2008 7:23pm
    #1
    L'prof
    Registered Users, Registered Users 2 Posts: 12,046 ✭✭✭✭


    Hi, I'm studying for exams (last day of study week) and one of the questions that comes up is on curvature. So I need to Paramaterize the curve first...thing is I'm not sure how! I know it involves the sin and cos functions...

    y = x^2

    5x^2 + 4y^2 = 9

    x^2 + xy + y^2 = 7

    These are the examples from my problem sheet, I know how to find the curvature but not really sure how to paramaterize them!

    I think I can paramaterize the second one:

    x = 3cos(t)/sqrt(5)
    y = 3sin(t)/2


Comments

  • #2
    Michael Collins
    Moderators, Science, Health & Environment Moderators Posts: 1,852 Mod ✭✭✭✭Michael Collins


    You have it right for the second one. The first one is actually easier:

    x = t
    y = t^2

    For the third one it's a conic section anyway, but I can't think of a paramaterization off the top of my head. Maybe there's a similar example in your notes?


  • #3
    L'prof
    Registered Users, Registered Users 2 Posts: 12,046 ✭✭✭✭L'prof


    Michael Collins wrote: »
    You have it right for the second one. The first one is actually easier:

    x = t
    y = t^2

    For the third one it's a conic section anyway, but I can't think of a paramaterization off the top of my head. Maybe there's a similar example in your notes?

    How do you get x = t, y = t^2 for the first one? I missed that lecture (or two) so I have no notes on it! It's not coming up as a full question, but I'd like to be able to do it!


  • #4
    Michael Collins
    Moderators, Science, Health & Environment Moderators Posts: 1,852 Mod ✭✭✭✭Michael Collins


    It's very easy to get the parameterization when you can get one variable in terms of the other. In this case you've given y in terms of x. So in that case you can just set

    x = t
    y = x^2 = t^2

    If it was y = x^2 + 5x + 2 it would still be simple. The difficulty arises when you can't seperate variables so easy, like in your third example there - you can't get y in terms of x in that one.


  • #5
    L'prof
    Registered Users, Registered Users 2 Posts: 12,046 ✭✭✭✭L'prof


    Michael Collins wrote: »
    It's very easy to get the parameterization when you can get one variable in terms of the other. In this case you've given y in terms of x. So in that case you can just set

    x = t
    y = x^2 = t^2

    If it was y = x^2 + 5x + 2 it would still be simple. The difficulty arises when you can't seperate variables so easy, like in your third example there - you can't get y in terms of x in that one.

    Ah OK, I just didn't understand that that was the method! I'm studying it tomorrow for the exam on Tuesday...I'll figure out how to work it out and post here again. Thanks for your help :)


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