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Math Help - At what point does this parametric curve cross itself?

  1. #1
    s3a
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    At what point does this parametric curve cross itself?

    Problem:The parametric curve r = (–t^2 – 4t – 4, –3cos(πt), t^3 – 28t) crosses itself at one and only one point P.a) What are the coordinates of this point P?b) Letting θ be the acute angle between the two tangent lines to the curve at the crossing point, what is the value of cos(θ)?Answers:a) P(–16,–3,–48)b) cos(θ) = 0.934487734928968I'm trying to follow this link ( Pauls Online Notes : Calculus II - Tangents with Parametric Equations ), but I'm having trouble adapting that to this particular problem.Could someone please tell me how to go about solving this problem?
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    Re: At what point does this parametric curve cross itself?

    Quote Originally Posted by s3a View Post
    Problem:The parametric curve r = (–t^2 – 4t – 4, –3cos(πt), t^3 – 28t) crosses itself at one and only one point P.a) What are the coordinates of this point P?b)
    I can tell you that the most difficult part of this problem is finding the point of self intersection.
    Find t_1\ne t_2 such that r(t_1)=r(t_2). I frankly see no clear solution to that.

    Finding the angle is easy: \theta  = \arccos \left( {\frac{{ {r'({t_1}) \cdot r'({t_2})} }}{{\left\| {r'({t_1})} \right\|\,\left\| {r'({t_2})} \right\|}}} \right)
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    Re: At what point does this parametric curve cross itself?

    If the point of intersection has parameters t_{1} and t_{2}, then

    -t_{1}^{2}-4t_{1}-4 = -t_{2}^{2}-4t_{2}-4,

    and

    t_{1}^{3}-28t_{1}=t_{2}^{3}-28t_{2}.

    Solve simultaneously to get the values -6 and 2.
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    s3a
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    Re: At what point does this parametric curve cross itself?

    Thanks for the replies, guys!

    BobP, I'm a little too sleepy to do this manually now, so I used software and got t_1 = -2, t_1 = 2, t_1 = -6, as well as t_2 = -2, t_2 = 2, t_2 = -6. How do we know to reject t_2 = -2?
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    Re: At what point does this parametric curve cross itself?

    I don't know why your software came up with this -2. I did this on paper and found just the two values, -6 and +2.
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    Re: At what point does this parametric curve cross itself?

    OK., it's because it's accepting t_{1}=t_{2} as a possible solution.
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    s3a
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    Re: At what point does this parametric curve cross itself?

    Hello, again.

    That answer doesn't sit well with me, because, in my work (which is attached in this post), I get t_2 = –2 by examining t_1 = –t_2 – 4 and not t_1 = t_2.

    If I examine t_1 = t_2, then I get the set of all solutions (which makes sense to me).

    Having said that, t_2 = –2 IS wrong (because it doesn't give the correct answer when I plug it into r), but can you help me see why? Am I supposed to plug in all points, and reject what disagrees with the majority, or is there a better way than that?

    Could you also tell me why finding t_2 ≠ t_2 such that r(t_1) = r(t_2) yields the final answer I'm looking for, in the first place?
    Attached Thumbnails Attached Thumbnails At what point does this parametric curve cross itself?-mywork.jpg  
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    Re: At what point does this parametric curve cross itself?

    You've arrived at the correct result haven't you ?

    I agree with your first equation t_{1}=-t_{2}-4, and if you substitute the first of the three possible values you have at the bottom of the page t_{2}=-2, you find that t_{1}=-2 also. So, you reject that since the two values cannot be the same. That leaves you with the other two.

    My algebra is different from yours.

    That first equation can be written as

    t_{2}^{2}-t_{1}^{2}+4t_{2}-4t_{1}=0, so

    (t_{2}-t_{1})(t_{2}+t_{1})+4({t_{2}-t_{1})=0,

    (t_{2}-t_{1})(t_{2}+t_{1}+4)=0

    and rejecting t_{2}=t_{1},

    we have t_{2}+t_{1}+4=0.

    The same technique can be used for the second equation.
    It needs the identity

    a^{3}-b^{3}=(a-b)(a^{2}+ ab +b^{2}).

    Then substitute from the first into the second to arrive at a quadratic.
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    s3a
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    Re: At what point does this parametric curve cross itself?

    Yes, I have arrived at the correct answer.

    However, I have arrived at the correct answer by just assuming what Plato said, without understanding why that is the case.

    To be specific, I made a typo in my previous post when I asked “Could you also tell me why finding t_2 ≠ t_2 such that r(t_1) = r(t_2) yields the final answer I'm looking for, in the first place?”.; I meant to ask “Could you also tell me why finding t_1 ≠ t_2 such that r(t_1) = r(t_2) yields the final answer I'm looking for, in the first place?”

    Basically, I now know to reject t_2 = –2, because I don't want t_1 to equal to t_2, but what is the specific, formal reasoning for not being allowed to have t_1 equal to t_2, and why does finding t_1 and t_2 such that t_1 ≠ t_2 and r(t_1) = r(t_2) allow us to find the point P?
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    Re: At what point does this parametric curve cross itself?

    Quote Originally Posted by s3a View Post
    I meant to ask “Could you also tell me why finding t_1 ≠ t_2 such that r(t_1) = r(t_2) yields the final answer I'm looking for, in the first place?”
    For a curve to self-intersect means the actual graph must "loop".
    If one thinks of t as a time variable, then the plot is in the exact same place at different times..
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  11. #11
    s3a
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    Re: At what point does this parametric curve cross itself?

    Oh, yes, I now get part a!

    For part b, however, I get dr(t)/dt = (-2t – 4, 3πsin(πt), 3t^2 – 28)

    dr(-6)/dt = (8, 0, 80) = r'(-6)

    dr(2)/dt = (-8,0,-4) = r'(2)

    cos(θ) = [r'(-6) ⋅ r'(2)]/[||r'(-6)|| ||r'(2)||] = -384/[sqrt(80) * sqrt(6464)] = -0.53399299138798174728
    ≠ 0.934487734928968
    (which is the correct final answer).

    What am I doing wrong?
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  12. #12
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    Re: At what point does this parametric curve cross itself?

    You've just made one simple mistake in your arithmetic.
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  13. #13
    s3a
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    Re: At what point does this parametric curve cross itself?

    I think I found the mistake you speak of.:
    r'(2) = (-8,0,-16) and not (-8,0,-4).

    I get the rest.

    Thank you both very much!
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    Re: At what point does this parametric curve cross itself?

    Quote Originally Posted by s3a View Post
    I think I found the mistake you speak of.:
    r'(2) = (-8,0,-16) and not (-8,0,-4).
    I get the rest.
    I reread the OP.
    To get the acute angle we need to use absolute value |R'(t_1)\cdot R'(t_2)|
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  15. #15
    s3a
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    Re: At what point does this parametric curve cross itself?

    Actually, I found the angle that was roughly 160 degrees, and I converted it to the angle that was roughly 20 degrees, which yields the same answer, but I prefer my way, because it makes me understand what is going on geometrically a little better, but thank you for emphasizing that the answer must be positive. (When drawing the tangents, there are two pairs of angles, where each pair consists of two equivalent angles, and I chose (one of) the acute one(s).)
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