Arc Length and Curvature

This guide explores how to quantify the distance along a path in space and measure the "sharpness" of its turns using the concepts of arc length and curvature.

Learning Objectives

After studying this material, you should be able to:

• Calculate the arc length of both plane and space curves using vector-valued functions.
• Derive the arc-length function $s(t)$ and use it to find an arc-length parameterization.
• Define and calculate the curvature $\kappa$ of a smooth curve.
• Understand the geometric significance of the Unit Tangent, Normal, and Binormal vectors.

Key Terms & Glossary

• Smooth Curve: A curve parameterized by $\mathbf{r}(t)$ where $\mathbf{r}'(t)$ is continuous and $\mathbf{r}'(t) \neq \mathbf{0}$ for all $t$.
  • Example: A circle is smooth; a curve with a "cusp" (like $y^2 = x^3$) is not smooth at the origin.
• Arc-Length Parameter ($s$): A parameter that represents the actual distance traveled along a curve from a starting point.
• Curvature ($\kappa$): A scalar measure of how quickly a curve changes direction at a point.
  • Example: A circle of radius $R$ has a constant curvature of $1/R$. A straight line has 0 curvature.
• Osculating Circle: The "kissing circle" that best approximates a curve at a specific point.

The "Big Idea"

In basic physics, we often describe motion in terms of time ($t$). However, the geometry of a path doesn't care how fast you move; it only cares about the shape. By switching from a time parameter ($t$) to an arc-length parameter ($s$), we describe the curve based on its internal geometry. This allows us to define curvature, which tells us how much the unit tangent vector is "twisting" per unit of distance traveled.

Formula / Concept Box

Hierarchical Outline

1. Arc Length Calculation
   • Plane Curves: $$\int \sqrt{[x'(t)]^2 + [y'(t)]^2}$dt$
   • Space Curves: $$\int \sqrt{[x'(t)]^2 + [y'(t)]^2 + [z'(t)]^2}$dt$
2. Arc-Length Parameterization
   • Step 1: Find $s(t)$ by integrating speed.
   • Step 2: Solve the resulting equation for $t$ in terms of $s$.
   • Step 3: Substitute $t(s)$ back into $\mathbf{r}(t)$.
3. Curvature Analysis
   • Relation to radius: $\kappa = 1/R$.
   • Smoothness requirement: $$\mathbf{r}$'(t)$ must not be zero to avoid division by zero.

Visual Anchors

Parameterization Logic

The TNB Frame

Definition-Example Pairs

[!IMPORTANT] Definition: Arc-Length Parameterization A curve $\mathbf{r}(s)$ is parameterized by arc length if the speed \|\mathbf{r}$'(s)\| = 1$ for all $s$.

• Example: Consider \mathbf{r}(t) = \langle 3t$, $4t \rangle.
  • \mathbf{r}$'(t) $= \langle 3$, $4 \rangle, so speed is $\sqrt{3^2+4^2} = 5$.
  • $s(t)$= \int_0^t 5$du = 5t$.
  • Solving for $t$: $t = s/5$.
  • Arc-length version: \mathbf{r}(s) = \langle 3(s/5)$, 4(s/5) $\rangle = \langle \frac{3}{5}s$, $\frac{4}{5}s \rangle.
  • Check: Velocity is \langle 3/5$, $4/5 \rangle, magnitude is 1. Success!

Worked Examples

Example: Arc Length of a Helix

Problem: Find the arc length of the helix \mathbf{r}(t) = \langle \cos t$, $\sin t$, $t \rangle from $t=0$ to $t=2\pi$.

Solution:

1. Find Derivative: \mathbf{r}$'(t) $= \langle -\sin t$, $\cos t$, $1 \rangle.
2. Calculate Magnitude (Speed): $\|\mathbf{r}'(t)\| = \sqrt{(-\sin t)^2 + (\cos t)^2 + 1^2} = \sqrt{\sin^2 t + \cos^2 t + 1} = \sqrt{1 + 1} = \sqrt{2}$
3. Integrate: $L = \int_0^{2\pi} \sqrt{2} dt = \left[ \sqrt{2}t \right]_0^{2\pi} = 2\pi\sqrt{2}$

[!TIP] For a helix, the speed is constant, making the integration very straightforward!

Checkpoint Questions

1. If a curve has a speed of constant 1, what does the parameter $t$ represent?
2. Why must a curve be "smooth" to calculate curvature using the standard formulas?
3. What is the curvature of a circle with a diameter of 10 units?
4. Solve the relationship: if $s(t) = 3t^2$, what is the speed of the particle at time $t=2$?

▶Click to view answers

1. The parameter $t$ represents the arc length $s$.
2. Smoothness ensures the derivative is non-zero, preventing division by zero in the curvature formula.
3. $\kappa = 1/R$. If diameter is 10, $R=5$, so $\kappa = 1/5 = 0.2$.
4. Speed $v(t)$= \frac{ds}{dt} = 6t$$. At $t=2$, speed is 12 units/sec.