Calc 3 (parametric curve corssing itself)?

Question

The parametric curve r = (t^2 + 4t - 3, -cos(pi*t), t^3 - 37t) crosses itslef at one and only one point. The point is (x,y,z) = ?

Let theta br the acute angle between th two tangent lines to the curve at the crossing point.The cos(theta) = ?

Show Work Please!

Answer 1

For the curve to cross itself, there must be t₁ and t₂ such that r(t₁)=r(t₂) but t₁≠t₂. In particular, this means r_x(t₁)=r_x(t₂) so that:

t₁² + 4t₁ - 3 = t₂² + 4t₂ - 3

Solving for t₁:

t₁² + 4t₁ - 3 = t₂² + 4t₂ - 3
t₁² + 4t₁ + 4 = t₂² + 4t₂ + 4
(t₁+2)² = (t₂+2)²
t₁+2 = ±(t₂+2)

But since we want the solution where t₁≠t₂, we take the minus sign to obtain:

t₁+2 = -t₂-2
t₁=-t₂-4

Or equivalently:

t₂=-t₁-4

Thus, if r(t) is the point of intersection, then r(t) = r(-t-4). So in particular, r_z(t) = r_z(-t-4). This can considerably narrow our search, substituting reveals that:

t³-37t = (-t-4)³ - 37(-t-4)

Expanding:

t³-37t = -t³ - 12t² - 48t - 64 + 37t + 148

Simplifying:

t³ - 37t = -t³ - 12t² - 11t + 84

Adding:

2t³ + 12t² - 26t - 84 = 0

Dividing by 2:

t³ + 6t² - 13t - 42 = 0

Skipping the root testing, this polynomial has roots at t=-2, t=-7, and t=3. So the only points where the curve could possibly cross itself are r(-2), r(-7), and r(3). All that remains is to test this finite set of points:

r(-2) = (-7, -1, 66)
r(-7) = (18, 1, -84)
r(3) = (18, 1, -84)

So the point where it crosses itself is r(3) = r(-7) = (18, 1, -84)

Now, to find the angle between the two tangent lines to the curve, first we actually find the tangent vectors:

r'(t) = (2t +4, π sin (πt), 3t² - 37)
r'(-7) = (-10, 0, 110)
r'(3) = (10, 0, -10)

Now, the cosine of the angle between the tangent vectors is r'(-7)·r'(3)/(||r'(-7)||*||r'(3)||), which is:

(-100-1100)/(√(100+12100)√(100+100))
-1200/(√12200√200)
-1200/(100√122√2)
-12/(√244)
-6/√61

But we are looking for the acute angle, and this being a negative number is the cosine of the obtuse angle. The acute angle between the tangent lines is then the supplement of the angle. So we have:

-6/√61 = cos (π-θ) = -cos θ
cos θ = 6/√61
θ≈39.8°

And we are done.