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Chapter 12: Problem 7

Explain what it means for a curve to be parameterized by its arc length.

Short Answer

Expert verified
Answer: Parameterizing a curve by its arc length offers advantages when working with geometric properties of the curve, such as calculating curvature and torsion. The computations are usually less complex in this form, as the magnitude of the curve's derivative is 1, simplifying many formulas. Additionally, this parameterization removes the dependency on any specific coordinate system, emphasizing the curve's intrinsic properties.

Step by step solution

01

Definition of a Parameterized Curve

A parameterized curve is a curve represented in terms of a parameter, usually denoted by \(t\). A two-dimensional parameterized curve can be expressed as \(\textbf{r}(t) = \langle x(t), y(t) \rangle\), where \(x(t)\) and \(y(t)\) are functions of the parameter \(t\). A three-dimensional curve can be represented similarly, with the addition of the component \(z(t)\).
02

Definition of Arc Length

Arc length is the length of a curve, usually denoted by \(s\). For a smooth parameterized curve \(\textbf{r}(t)\) in two or three dimensions, where \(t \in [a, b]\), the arc length \(s\) from the point \(\textbf{r}(a)\) to \(\textbf{r}(b)\) can be calculated as: \[s = \int_{a}^{b} |\textbf{r}'(t)| dt\]
03

Parameterization by Arc Length

A curve is said to be parameterized by arc length if the parameter, in this case, \(t\), represents the arc length \(s\) from a fixed point on the curve to the point \(\textbf{r}(t)\), for every \(t\). This means that the parameter \(t\) increases at a constant rate with respect to the distance traveled along the curve. In this case, the arc length formula simplifies to:\[|\textbf{r}'(t)| = ds/dt = 1\].
04

Advantages of Arc Length Parameterization

Parameterizing a curve by arc length has advantages when working with geometric properties of the curve such as calculating curvature and torsion. The computations are usually less complex in this form since the magnitude of the curve's derivative is 1, simplifying many formulas. Furthermore, this parameterization eliminates the dependency on any specific coordinate system, emphasizing the curve's intrinsic properties.

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Most popular questions from this chapter

The definition \(\mathbf{u} \cdot \mathbf{v}=|\mathbf{u}||\mathbf{v}| \cos \theta\) implies that \(|\mathbf{u} \cdot \mathbf{v}| \leq|\mathbf{u}||\mathbf{v}|(\text {because}|\cos \theta| \leq 1) .\) This inequality, known as the Cauchy-Schwarz Inequality, holds in any number of dimensions and has many consequences. Use the vectors \(\mathbf{u}=\langle\sqrt{a}, \sqrt{b}\rangle\) and \(\mathbf{v}=\langle\sqrt{b}, \sqrt{a}\rangle\) to show that \(\sqrt{a b} \leq(a+b) / 2,\) where \(a \geq 0\) and \(b \geq 0\).

For the given points \(P, Q,\) and \(R,\) find the approximate measurements of the angles of \(\triangle P Q R\). $$P(1,-4), Q(2,7), R(-2,2)$$

Find the domains of the following vector-valued functions. $$\mathbf{r}(t)=\sqrt{4-t^{2}} \mathbf{i}+\sqrt{t} \mathbf{j}-\frac{2}{\sqrt{1+t}} \mathbf{k}$$

Consider the curve \(\mathbf{r}(t)=(a \cos t+b \sin t) \mathbf{i}+(c \cos t+d \sin t) \mathbf{j}+(e \cos t+f \sin t) \mathbf{k}\) where \(a, b, c, d, e,\) and fare real numbers. It can be shown that this curve lies in a plane. Assuming the curve lies in a plane, show that it is a circle centered at the origin with radius \(R\) provided \(a^{2}+c^{2}+e^{2}=b^{2}+d^{2}+f^{2}=R^{2}\) and \(a b+c d+e f=0\).

A baseball leaves the hand of a pitcher 6 vertical feet above home plate and \(60 \mathrm{ft}\) from home plate. Assume that the coordinate axes are oriented as shown in the figure. a. In the absence of all forces except gravity, assume that a pitch is thrown with an initial velocity of \(\langle 130,0,-3\rangle \mathrm{ft} / \mathrm{s}\) (about \(90 \mathrm{mi} / \mathrm{hr}\) ). How far above the ground is the ball when it crosses home plate and how long does it take for the pitch to arrive? b. What vertical velocity component should the pitcher use so that the pitch crosses home plate exactly \(3 \mathrm{ft}\) above the ground? c. A simple model to describe the curve of a baseball assumes that the spin of the ball produces a constant sideways acceleration (in the \(y\) -direction) of \(c \mathrm{ft} / \mathrm{s}^{2}\). Assume a pitcher throws a curve ball with \(c=8 \mathrm{ft} / \mathrm{s}^{2}\) (one-fourth the acceleration of gravity). How far does the ball move in the \(y\) -direction by the time it reaches home plate, assuming an initial velocity of (130,0,-3) ft/s? d. In part (c), does the ball curve more in the first half of its trip to the plate or in the second half? How does this fact affect the batter? e. Suppose the pitcher releases the ball from an initial position of (0,-3,6) with initial velocity \((130,0,-3) .\) What value of the spin parameter \(c\) is needed to put the ball over home plate passing through the point (60,0,3)\(?\)
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