PGDDSA Study · Semester 1

Core Titles

Key headlines and terms for quick recall

Arc length $L = \int_a^b \|\vec{r}'(t)\| \, dt$
Arc-length parameter $s(t) = \int_a^t \|\vec{r}'(u)\| \, du$
Unit tangent $\vec{T}(t) = \vec{r}'(t)/\|\vec{r}'(t)\|$
Curvature $\kappa(t) = \dfrac{\|\vec{T}'(t)\|}{\|\vec{r}'(t)\|} = \dfrac{\|\vec{r}'(t) \times \vec{r}''(t)\|}{\|\vec{r}'(t)\|^3}$
Radius of curvature $R = 1/\kappa$

Basic Idea

What it is, why it matters, how it works

Arc length

The length of the curve $\vec{r}(t) = \langle f(t), g(t), h(t) \rangle$ for $a \le t \le b$ is: $L = \int_a^b \|\vec{r}'(t)\| \, dt = \int_a^b \sqrt{f'^2 + g'^2 + h'^2} \, dt.$ Why? Approximate the curve by small straight segments. Each has length $\|\vec{r}(t + \Delta t) - \vec{r}(t)\| \approx \|\vec{r}'(t)\| \Delta t$ . Sum and pass to the integral.

Arc-length parameter

$s(t) = \int_a^t \|\vec{r}'(u)\| \, du.$ A reparametrisation by $s$ traces the curve at unit speed: $\|\vec{r}'(s)\| = 1$ .

Unit tangent

$\vec{T}(t) = \frac{\vec{r}'(t)}{\|\vec{r}'(t)\|}.$

Curvature

Curvature $\kappa$ measures how sharply the curve is bending — the rate at which the unit tangent changes direction per unit arc length. $\kappa = \left\| \frac{d\vec{T}}{ds} \right\| = \frac{\|\vec{T}'(t)\|}{\|\vec{r}'(t)\|}.$

A handy alternative formula: $\kappa(t) = \frac{\|\vec{r}'(t) \times \vec{r}''(t)\|}{\|\vec{r}'(t)\|^3}.$

For a plane curve $y = f(x)$ : $\kappa = \frac{|f''(x)|}{(1 + f'(x)^2)^{3/2}}.$

Radius of curvature

$R = \frac{1}{\kappa}$ — the radius of the circle that best approximates the curve locally (osculating circle).

Why this matters in Data Science

Manifold learning: estimating curvature of data manifolds.
Motion planning: minimising path length and curvature in robotics.
Information-geometry distances on statistical manifolds.

Mind Map

Visual structure of the concept

ARC LENGTH & CURVATURE
├── L = ∫ₐᵇ ‖r'(t)‖ dt
├── Arc-length parameter s(t) = ∫ ‖r'(u)‖ du
├── Unit tangent T(t) = r'/‖r'‖
├── Curvature κ
│   ├── κ = ‖dT/ds‖
│   ├── κ = ‖r'×r''‖ / ‖r'‖³
│   └── Plane y=f(x): κ = |f''| / (1 + f'²)^{3/2}
└── Radius of curvature R = 1 / κ

Exam Q&A

Part A (2 marks) and Part B (20 marks) style questions

Part A (2 marks each)

Q1. Write the formula for arc length of $\vec{r}(t)$ on $[a, b]$ . $L = \int_a^b \|\vec{r}'(t)\| \, dt$ .

Q2. Define curvature. The magnitude of the rate of change of the unit tangent with respect to arc length: $\kappa = \|d\vec{T}/ds\|$ .

Q3. Find arc length of $\vec{r}(t) = (\cos t, \sin t, t)$ on $[0, 2\pi]$ . $\|\vec{r}'\| = \sqrt{2}$ , $L = 2\pi\sqrt{2}$ .

Q4. State the formula for curvature using $\vec{r}'$ and $\vec{r}''$ . $\kappa = \dfrac{\|\vec{r}' \times \vec{r}''\|}{\|\vec{r}'\|^3}$ .

Part B (20 marks)

Q. Derive the formula for arc length of a curve $\vec{r}(t)$ . Define curvature and derive both forms — $\kappa = \|d\vec{T}/ds\|$ and $\kappa = \|\vec{r}' \times \vec{r}''\| / \|\vec{r}'\|^3$ . Compute the arc length and curvature of the helix $\vec{r}(t) = (a\cos t, a\sin t, bt)$ .

Arc length. Approximate the curve by short chords. For a small change $\Delta t$ : $\vec{r}(t + \Delta t) - \vec{r}(t) \approx \vec{r}'(t) \Delta t,$ so chord length $\approx \|\vec{r}'(t)\| \Delta t$ . Summing and taking the limit: $L = \int_a^b \|\vec{r}'(t)\| \, dt.$

Arc-length parameter. Define $s(t) = \int_a^t \|\vec{r}'(u)\| \, du$ . Then $\dfrac{ds}{dt} = \|\vec{r}'(t)\|$ , so reparametrising by $s$ gives unit speed.

Curvature.

First formula. The unit tangent is $\vec{T} = \vec{r}'/\|\vec{r}'\|$ . Curvature is the rate at which $\vec{T}$ turns per unit distance: $\kappa = \left\| \frac{d\vec{T}}{ds} \right\| = \frac{\|d\vec{T}/dt\|}{ds/dt} = \frac{\|\vec{T}'(t)\|}{\|\vec{r}'(t)\|}.$

Second formula. From $\vec{r}' = \|\vec{r}'\| \vec{T}$ , $\vec{r}'' = \|\vec{r}'\|' \vec{T} + \|\vec{r}'\| \vec{T}'.$ Cross with $\vec{r}'$ (using $\vec{T} \times \vec{T} = 0$ ): $\vec{r}' \times \vec{r}'' = \|\vec{r}'\|^2 \, \vec{T} \times \vec{T}'.$ Since $\vec{T}$ and $\vec{T}'$ are perpendicular (as $|\vec{T}|=1$ implies $\vec{T} \cdot \vec{T}' = 0$ ), $\|\vec{r}' \times \vec{r}''\| = \|\vec{r}'\|^2 \|\vec{T}'\| \Rightarrow \|\vec{T}'\| = \frac{\|\vec{r}' \times \vec{r}''\|}{\|\vec{r}'\|^2}.$ Hence $\kappa = \frac{\|\vec{T}'\|}{\|\vec{r}'\|} = \frac{\|\vec{r}' \times \vec{r}''\|}{\|\vec{r}'\|^3}. \quad \blacksquare$

Helix $\vec{r}(t) = (a\cos t, a\sin t, bt)$ for $t \in [0, 2\pi]$ .

$\vec{r}'(t) = (-a\sin t, a\cos t, b)$ , $\|\vec{r}'\| = \sqrt{a^2 + b^2}$ (constant).

Arc length: $L = \int_0^{2\pi} \sqrt{a^2 + b^2} \, dt = 2\pi\sqrt{a^2 + b^2}$ .

Curvature. $\vec{r}'' = (-a\cos t, -a\sin t, 0)$ . $\vec{r}' \times \vec{r}'' = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ -a\sin t & a\cos t & b \\ -a\cos t & -a\sin t & 0 \end{vmatrix} = (ab\sin t,\, -ab\cos t,\, a^2)$ (after simplification). Magnitude: $\sqrt{a^2 b^2 + a^4} = a\sqrt{a^2 + b^2}$ .

$\kappa = \frac{a\sqrt{a^2 + b^2}}{(a^2 + b^2)^{3/2}} = \frac{a}{a^2 + b^2}.$

The helix has constant curvature — a defining property — and radius of curvature $R = (a^2 + b^2)/a$ .

Arc Length and Curvature