The helix r(t) = ⟨cos t, sin t, t⟩ has derivative r'(t) = ⟨−sin t, cos t, 1⟩. What is the arc length from t = 0 to t = 2π?
A2π, the same as a flat unit circle of radius 1
B√2, the speed at any point along the helix
C2π√2, found by integrating the constant speed √2 over the interval [0, 2π]
D2π + 2π = 4π, adding the circular length and the vertical rise separately
|r'(t)| = √(sin²t + cos²t + 1) = √(1 + 1) = √2, which is constant. The arc length is ∫₀^{2π} √2 dt = 2π√2. This is longer than a flat circle (2π) by the factor √2, which accounts for the simultaneous upward climb. Adding lengths independently (option D) is not valid — the speed already captures both the circular and vertical components at once through the magnitude of the full velocity vector.
Question 2 Multiple Choice
A curve r(t) traces a helix for t ∈ [0, 2π]. If you reparametrize it as r(2s) for s ∈ [0, π], which traces the same curve twice as fast, how does the arc length change?
AThe arc length halves, because the integration interval [0, π] is half as long
BThe arc length stays the same, because arc length is a property of the curve as a geometric object, independent of how it is parametrized
CThe arc length doubles, because the speed doubles under the reparametrization
DThe arc length changes unpredictably and must be recomputed from scratch
Arc length is parametrization-independent. Under reparametrization r(2s), the speed |dr/ds| = 2|r'(2s)| doubles, but the integration interval is half as long — [0, π] instead of [0, 2π]. These effects cancel exactly, giving the same arc length. This is not a coincidence: arc length is defined to capture a geometric property of the curve (the distance traveled), which cannot depend on whether you traverse it quickly or slowly.
Question 3 True / False
Arc length of a parametric curve r(t) from a to b equals the integral of the speed |r'(t)| over [a, b].
TTrue
FFalse
Answer: True
This is the fundamental formula: L = ∫ₐᵇ |r'(t)| dt. The velocity vector r'(t) gives the direction and speed of travel along the curve; its magnitude |r'(t)| is the instantaneous speed. Integrating speed over time gives total distance — the same principle as distance = ∫ speed dt in one dimension, generalized to curves winding through 3D space.
Question 4 True / False
Reparametrizing a curve changes its arc length, because the derivative r'(t) changes when you substitute a new parameter.
TTrue
FFalse
Answer: False
Arc length is a geometric invariant — it measures the physical length of the curve as a set of points in space, which cannot depend on the parameter used to trace it. A reparametrization changes both the integrand (the speed |r'|) and the integration limits in compensating ways: faster traversal over a shorter interval yields the same total length. This is why arc length is called a 'parametrization-independent' quantity and why the arc-length parametrization (where speed = 1 everywhere) is considered the 'natural' one.
Question 5 Short Answer
Explain why arc length is described as 'parametrization-independent,' and why this property motivates the definition of arc-length parametrization.
Think about your answer, then reveal below.
Model answer: Arc length measures the physical distance along the curve as a geometric object — the distance doesn't change just because you choose to traverse it faster or slower. Formally, any reparametrization t = φ(s) changes both |r'| and the integration limits such that the product remains the same. This geometric invariance motivates the arc-length parametrization s, defined so that |r'(s)| = 1 everywhere: moving along the curve at unit speed. Under this parametrization, the arc length from s₀ to s₁ is simply s₁ − s₀, simplifying formulas for curvature and torsion.
The arc-length parameter s is the 'natural' parametrization because it removes the arbitrary choice of traversal speed from all geometric calculations. Curvature κ = |r''(s)| under arc-length parametrization, for example, measures only how the curve bends — not how fast you happen to move along it. Without parametrization independence, geometric properties of curves would depend on how you describe them rather than on the curves themselves.