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Points of intersection with regularly parametrized curves

Hello,

How would I go about finding the points of intersection between the parametrized curve γ\gamma = (t³, t²-t) and its tangent line vector at t=-1: γ^\hat{\gamma} = (-1+t, 2-t)?

Any help to get me started on this question would be most appreciated :smile:
(edited 9 years ago)
Reply 1
Avatar for octionut
octionut
2
Are you sure the tangent line vector is correct?
Reply 2
Avatar for razzor
razzor
OP
10
It is what I had calculated. I will send you the exact question which is in 3 parts:

Let γ:\gamma: ℝ -> ℝ² such that γ(t)=(t3,t2−t)\gamma (t) = (t^{3}, t^{2}-t)

a) Show that γ\gamma is a regularly parametrized curve.

b) Construct its tangent line at time t = -1, γ^−1\hat{\gamma}_{-1} : ℝ -> ℝ²
c) Find all intersection points betwen γ\gamma and γ^−1\hat{\gamma}_{-1}

For part b), I found that γ^−1(t)=(−1,2)+t(3,−3)\hat{\gamma}_{-1}(t) = (-1,2) + t(3,-3)
or reparametrizing: γ^−1(t)=(−1,2)+t(1,−1)\hat{\gamma}_{-1}(t) = (-1,2) + t(1,-1)

That's how I obtained the tangent line vector
(edited 9 years ago)
Reply 3
Avatar for octionut
octionut
2
Original post by razzor
It is what I had calculated. I will send you the exact question which is in 3 parts:

Let γ:\gamma: ℝ -> ℝ² such that γ(t)=(t3,t2−t)\gamma (t) = (t^{3}, t^{2}-t)

a) Show that γ\gamma is a regularly parametrized curve.

b) Construct its tangent line at time t = -1, γ^−1\hat{\gamma}_{-1} : ℝ -> ℝ²
c) Find all intersection points betwen γ\gamma and γ^−1\hat{\gamma}_{-1}

For part b), I found that γ^−1(t)=(−1,2)+t(3,−3)\hat{\gamma}_{-1}(t) = (-1,2) + t(3,-3)
or reparametrizing: γ^−1(t)=(−1,2)+t(1,−1)\hat{\gamma}_{-1}(t) = (-1,2) + t(1,-1)

That's how I obtained the tangent line vector


I may look like a fool now but "γ:\gamma: ℝ -> ℝ² such that γ(t)=(t3,t2−t)\gamma (t) = (t^{3}, t^{2}-t)" is just a parametric curve with x=t3 and y=t2-t, right? If that's the case, I found the tangent line at t=-1 to be -y=2/3+(2/3)x and that's not the same as x=-1+t, y=2-t.

Sorry for that, I might have misunderstood the question as my highest maths qualification is an A Level in Further Maths. Just ignore me if I'm wrong. :colondollar:
Reply 4
Avatar for razzor
razzor
OP
10
Original post by octionut
I may look like a fool now but "γ:\gamma: ℝ -> ℝ² such that γ(t)=(t3,t2−t)\gamma (t) = (t^{3}, t^{2}-t)" is just a parametric curve with x=t3 and y=t2-t, right? If that's the case, I found the tangent line at t=-1 to be -y=2/3+(2/3)x and that's not the same as x=-1+t, y=2-t.

Sorry for that, I might have misunderstood the question as my highest maths qualification is an A Level in Further Maths. Just ignore me if I'm wrong. :colondollar:



The tangent line is meant to be expressed in parametric form.

Don't worry, this is degree level :wink:
I think I have a vague idea on how to tackle this question, but I'm not 100% sure
Reply 5
Avatar for octionut
octionut
2
Original post by razzor
The tangent line is meant to be expressed in parametric form.

Don't worry, this is degree level :wink:
I think I have a vague idea on how to tackle this question, but I'm not 100% sure

Still, if I didn't misunderstand the first part of the question, my cartesian-form tangent would be right and x=-1+t, y=2-t is just -y=x-1 which would be wrong; but sorry again, I'll just stop wasting your time. :colondollar:
Reply 6
Avatar for razzor
razzor
OP
10
No that's fine! Thanks anyway for trying to help :smile:

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