Week 7 Activity

Section 7.2 Week 7 Activity

Subsection 7.2.1 Tangents to Parametric Curves

Activity 7.2.1.

Calculate the tangent to this parametric curve. (This is the same curve from Activity 6.3.1). Choose some points and draw some tangents to inspect if they fit the curve.

\begin{align*} \amp \gamma(t) = (t^2, t^3) \amp \amp t \in [-3,3] \end{align*}

Solution.

The tangent is the derivative calculated termwise.

\begin{equation*} \gamma^\prime(t) = (2t, 3t^2) \end{equation*}

This parametric curve with some example tangents is shown in Figure 7.2.1

Figure 7.2.1. Tangents to \(\gamma(t) = (t^2, t^3) \)

Activity 7.2.2.

Calculate the tangent to this parametric curve. (This is the same curve from Activity 6.3.2. Choose some points and draw some tangents to inspect if they fit the curve.

\begin{align*} \amp \gamma(t) = (\cos t, 2 \sin t) \amp \amp t \in [0,2\pi] \end{align*}

Solution.

The tangent is the derivative calculated termwise.

\begin{equation*} \gamma^\prime(t) = (\sin t, - 2 \cos t) \end{equation*}

This parametric curve with some example tangents is shown in Figure 7.2.2

Figure 7.2.2. Tangents to \(\gamma(t) = (\cos t, 2 \sin t) \)

Activity 7.2.3.

Calculate the tangent to this parametric curve. (This is the same curve from Activity 6.3.3). Choose some points and draw some tangents to inspect if they fit the curve.

\begin{align*} \amp \gamma(t) = (4 \cos t + \sin 2t, 4 \sin t + \cos 2 t) \amp \amp t \in [0,2\pi] \end{align*}

Solution.

The tangent is the derivative calculated termwise.

\begin{equation*} \gamma^\prime(t) = (-4 \sin t + 2 \cos 2t, 4 \cos t - 2 \sin 2t) \end{equation*}

This parametric curve with some example tangents is shown in Figure 7.2.3

Figure 7.2.3. Tangents to \(\gamma(t) = (4 \cos t + \sin 2t, 4 \sin t + \cos 2t)\)

Activity 7.2.4.

Calculate the tangent to this parametric curve. (This is the same curve from Activity 6.3.4). Choose some points and draw some tangents to inspect if they fit the curve.

\begin{align*} \amp \gamma(t) = (3-t^2,t) \amp \amp t \in [-4,4] \end{align*}

Solution.

The tangent is the derivative calculated termwise.

\begin{equation*} \gamma^\prime(t) = (-t, 1) \end{equation*}

This parametric curve with some example tangents is shown in Figure 7.2.4

Figure 7.2.4. Tangents to \(\gamma(t) = (3-t^2,t)\)

Activity 7.2.5.

Calculate the tangent to this parametric curve. (This is the same curve from Activity 6.3.5). Choose some points and draw some tangents to inspect if they fit the curve.

\begin{align*} \amp \gamma(t) = \left( \frac{\pi^2}{t^2} \cos t, \frac{\pi^2}{t^2} \sin t \right) \amp \amp t \in [\pi, 4\pi] \end{align*}

Solution.

The tangent is the derivative calculated termwise.

\begin{equation*} \gamma^\prime(t) = \left( \frac{-2\pi^2}{t^3} \cos t - \frac{\pi^2}{t^2} \sin t, \frac{-2\pi^2}{t^3} \sin t + \frac{\pi^2}{t^2} \cos t \right) \end{equation*}

This parametric curve with some example tangents is shown in Figure 7.2.5

Figure 7.2.5. Tangents to \(\gamma(t) = \left( \frac{\pi^2}{t^2} \cos t, \frac{\pi^2}{t^2} \sin t \right) \)

Activity 7.2.6.

Calculate the tangent to this parametric curve. (This is the same curve from Activity 6.3.6). Choose some points and draw some tangents to inspect if they fit the curve.

\begin{align*} \amp \gamma(t) = (\sqrt{t}, \sqrt[3]{t} ) \amp \amp t \in [0,100] \end{align*}

Solution.

The tangent is the derivative calculated termwise.

\begin{equation*} \gamma^\prime(t) = \left( \frac{1}{2\sqrt{t}}, \frac{1}{3 \sqrt[3]{t^2}} \right) \end{equation*}

This parametric curve with some example tangents is shown in Figure 7.2.6.

Figure 7.2.6. Tangents to \(\gamma(t) = (\sqrt{t}, \sqrt[3]{t})\)

Subsection 7.2.2 Full Descriptions of Curves in \(\RR^3\)

Activity 7.2.7.

For this curve, calculate the speed, curvature, torsion. Also calculate the tangent vector, normal vector and binormal vector.

\begin{equation*} \gamma(t) = (t^3, 1 - t^2, 1 + t^2) \end{equation*}

Interpret the curve using the three scalars, focusing on the asymptotic behaviour. (The expressions get very hectic in this example -- it is very hard to construct examples where the functions don't quickly grow difficult. Use a computer to do the derivatives and cross product to save yourself some calculation. I used Wolfram Alpha for all the derivatives and cross products, and some of the algebraic simplifications as well. By the time you get to \(B^\prime\text{,}\) expect some complicated results.)

Solution.

I follow the lengthy process of calculating speed, curvature, torsion, the tangent vector, the normal vector, and the binormal vector.

\begin{align*} \gamma(t) \amp = (t^3, 1 - t^2, 1 + t^2) \\ \gamma^\prime(t) \amp = (x^\prime(t), y^\prime(t), z^\prime(t)) = (3t^2, -2t, 2t) \\ v(t) \amp = |\gamma^\prime(t)| = \sqrt{x^\prime(t)^2 + y^\prime(t)^2 + z^\prime(t)^2 }\\ \amp = \sqrt{ 9t^4 + 4t^2 + 4t^2} = t \sqrt{9t^2 + 4} \\ T(t) \amp = \frac{\gamma^\prime(t)}{|\gamma^\prime(t)|} = \frac{1}{t\sqrt{9t^2 + 4}} \left( 3t^2, -2t, 2t \right)\\ \amp = \left( \frac{3t}{\sqrt{9t^2+4}} \frac{-2}{\sqrt{9t^2+4}} \frac{2}{\sqrt{9t^2+4}} \right)\\ T^\prime(t) \amp = \left( \frac{12}{(9t^2 + 4)^{\frac{3}{2}}}, \frac{-18t}{(9t^2+4)^{\frac{3}{2}}}, \frac{18t}{(9t^2+4)^{\frac{3}{2}}} \right) \\ \amp = \frac{1}{(9t^2+4)^{\frac{3}{2}}} (12, -18t, 18t)\\ |T^\prime(t)| \amp = \frac{1}{(9t^2+4)^{\frac{3}{2}}} \sqrt{144 + 324t^2 + 324t^2} = \frac{1}{(9t^2+4)^{\frac{3}{2}}} \sqrt{144 + 648t^2}\\ \kappa(t) \amp = \frac{|T^\prime(t)|}{|\gamma^\prime(t)|} = \frac{\frac{1}{(9t^2+4)^{\frac{3}{2}}} \sqrt{ 144 + 648t^2}}{\sqrt{ 9t^4 + 4t^2}} = \frac{144 + 648 t^2}{(9t^2+4)^2}\\ N(t) \amp = \frac{T^\prime(t)}{|T^\prime(t)|} \frac{1}{\sqrt{144 + 648t^2}} \left( 12, -18t, 18t \right) \\ B(t) \amp = T(t) \times N(t) = \frac{1}{\sqrt{9t^2+4}} (3t, -2, -2) \times \frac{1}{\sqrt{144+684t^2}} (12, -18t, 18t)\\ \amp = \frac{1}{\sqrt{(9t^2+4)(144+648t^2)}} \left[ (3t, -2, -2) \times (12, -18t, 18t) \right] \\ \amp = \frac{1}{\sqrt{5832t^4 + 3888t^2 + 576}} (-72t, -24 - 54t^2, 24 - 54t^2)\\ B^\prime(t) \amp = \frac{1}{(81t^4 +54t^2 + 8)^{\frac{3}{2}}} \left( 6\sqrt{2} (81t^4 - 8), 9\sqrt{2} t(9t^2 + 4), -9\sqrt{2}t (63t^2 + 20) \right) \\ \tau(t) \amp = - \frac{B^\prime(t)}{|\gamma^\prime(t)|} \cdot N(t)\\ \amp = \frac{-\left( 12(6\sqrt{2}) (81t^4 - 8) - 18t(9\sqrt{2})(9t^2 + 4) - 18t(9\sqrt{2})(63t^2 + 20) \right)}{t\sqrt{(9t^2+4)(81t^4 + 54t^2 + 8)^3}}\\ \amp = \frac{-\sqrt{2}}{t\sqrt{(9t^2+4)(81t^4 + 54t^2 + 8)^3}} (5832t^4 - 11664t^2 - 4464) \end{align*}

As I warned you, this got pretty intense. However, I can simplify and gain useful information by just looking at the asymptotic order of the three scalars.

\begin{align*} v \amp \cong t^2 \\ \kappa \amp \cong \frac{1}{t^2} \\ \tau \amp \cong \frac{t^4}{t^9} = \frac{1}{t^5} \end{align*}

The speed is increasing. This makes sense, since the components are cubic and quadratic. The curve covers more distance as the inputs to these polynomials become large. The curvature starts out significant, incidating a curving shape at the start. However, asymptotically, the curvature drops to zero, making this approach a straight line. Something similar is true for torsion. Torsion is large when \(t\) is small, showing the twisting of the curve near the origin. Asymptotically, torsion decays very quickly, so the twisting quickly become insignificant as the curve straightens out.

Subsection 7.2.3 Conceptual Review Questions

What is a tangent to a parametric curve?
What is the relationship betwee arclength and tangents?
What are normals and binormals?
What is curvature?
What is the osculating plane and how does it relate to torsion?
How to tangents, normals and binormals (alternativley speed, curvature and torsion) completely describe the motion of curves in \(\RR^3\text{?}\)