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Week 5 Discussion Notes

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Arc Length

Here's a mnemonic to help you remember the formula for arc length: from calculus, you should recognize the "formula"

Δf≈f′(t) Δt.\Delta f \approx f'\p{t} \,\Delta t.

This still works for vector functions:

Δr⃗≈r⃗′(t) Δt  ⟹  Δs≈∥r⃗′(t)∥ Δt,\Delta \vec{r} \approx \vec{r}'\p{t} \,\Delta t \implies \Delta s \approx \norm{\vec{r}'\p{t}} \,\Delta t,

after taking magnitudes. Thus, this becomes

ds=∥r⃗′(t)∥ dt  ⟹  L=∫0Lds=∫ab∥r⃗′(t)∥ dt\diff{s} = \norm{\vec{r}'\p{t}} \,\diff{t} \implies L = \int_0^L \diff{s} = \int_a^b \norm{\vec{r}'\p{t}} \,\diff{t}

after integrating both sides. This should hopefully give a little intuition on why arc length is defined the way it is.

Arc Length Parametrization

A big source of confusion for me when I first learned the material is the difference between tt and ss. This is because when we write r⃗′(s)\vec{r}'\p{s}, we really mean dr⃗ds(s)\deriv{\vec{r}}{s}\p{s} and not dr⃗dt(s)\deriv{\vec{r}}{t}\p{s} (notice that the first derivative is taken with respect to arc length, ss, instead of tt).

The distinction is really important when doing certain calculations. For example, to calculate curvature, you have

κ=∥dT⃗ds∥≠∥dT⃗dt∥.\kappa = \norm{\deriv{\vec{T}}{s}} \neq \norm{\deriv{\vec{T}}{t}}.

If you want to use the tt-derivative to calculate arc length, then you will want to sue the chain rule instead:

κ=∥dT⃗ds∥=∥dT⃗dtdtds∥=∥dT⃗dt∥∥r⃗′∥.\kappa = \norm{\deriv{\vec{T}}{s}} = \norm{\deriv{\vec{T}}{t} \deriv{t}{s}} = \frac{\norm{\deriv{\vec{T}}{t}}}{\norm{\vec{r}'}}.