Unit 2: Curvature, Torsion & The Frenet-Serret Frame

At each point along a smooth curve $C$ in $\mathbb{R}^3$, we can attach a moving, right-handed orthonormal coordinate system known as the Frenet-Serret Moving Trihedron (or TNB Frame).

1. Construction of the TNB Frame

1. Unit Tangent Vector $\vec{T}(t)$:

Points in the instantaneous direction of motion:

2. Principal Unit Normal Vector $\vec{N}(t)$:

Because $|\vec{T}(t)| = 1$ is constant, $\vec{T}'(t)$ is orthogonal to $\vec{T}(t)$ ($\vec{T} \cdot \vec{T}' = 0$). The unit vector in this orthogonal direction is:

$\vec{N}(t)$ points directly in the direction that the curve is turning.

3. Binormal Unit Vector $\vec{B}(t)$:

Defined by the vector cross product to complete a right-handed orthonormal triad:

Because $\vec{T}$ and $\vec{N}$ are orthogonal unit vectors, $|\vec{B}| = |\vec{T}| |\vec{N}| \sin(\pi/2) = 1$, and $\vec{B}$ is perpendicular to both $\vec{T}$ and $\vec{N}$.

2. The Three Fundamental Osculating Planes

At any point $P$ on the curve, the three mutually orthogonal pairs of vectors define three fundamental planes:

1. The Osculating Plane (Spanned by $\vec{T}$ and $\vec{N}$):

• Normal Vector: $\vec{B}(t)$
• Equation: $(\vec{r} - \vec{r}_0) \cdot \vec{B} = 0$
• Geometric Meaning: The plane that comes closest to containing the curve locally; it contains the instantaneous circle of curvature.

2. The Normal Plane (Spanned by $\vec{N}$ and $\vec{B}$):

• Normal Vector: $\vec{T}(t)$
• Equation: $(\vec{r} - \vec{r}_0) \cdot \vec{T} = 0$
• Geometric Meaning: The plane orthogonal to the curve; all lines normal to the curve lie in this plane.

3. The Rectifying Plane (Spanned by $\vec{T}$ and $\vec{B}$):

• Normal Vector: $\vec{N}(t)$
• Equation: $(\vec{r} - \vec{r}_0) \cdot \vec{N} = 0$

1. Geometric Definition of Curvature $\kappa$

The curvature $\kappa$ of a smooth curve measures how rapidly the curve changes its direction per unit change in arc length:

Using the chain rule with parameter $t$:

2. General Parametric Curvature Formula in $\mathbb{R}^3$

Theorem: For any smooth space curve parameterized by $\vec{r}(t)$:

Proof:

Since $\vec{v}(t) = \vec{r}'(t) = v \vec{T}$ where $v = |\vec{r}'(t)| = \frac{ds}{dt}$: Differentiating velocity to get acceleration:

Since $\frac{d\vec{T}}{ds} = \kappa \vec{N} \implies \vec{T}'(t) = \frac{ds}{dt} \frac{d\vec{T}}{ds} = v \kappa \vec{N}$:

Now compute the cross product $\vec{r}'(t) \times \vec{r}''(t)$:

Since $\vec{T} \times \vec{T} = \vec{0}$ and $\vec{T} \times \vec{N} = \vec{B}$:

Taking the norm of both sides (since $|\vec{B}| = 1$):

Dividing by $|\vec{r}'|^3$:

3. Special Curvature Formulas

1. Plane Curve in Cartesian Form $y = f(x)$:

Parameterize as $\vec{r}(x) = \langle x, \; f(x), \; 0 \rangle$. Then $\vec{r}'(x) = \langle 1, \; y', \; 0 \rangle$ and $\vec{r}''(x) = \langle 0, \; y'', \; 0 \rangle$. $\vec{r}' \times \vec{r}'' = \langle 0, \; 0, \; y'' \rangle \implies |\vec{r}' \times \vec{r}''| = |y''|$. $|\vec{r}'| = \sqrt{1 + y'^2}$.

2. Plane Curve in Polar Coordinates $r = f(\theta)$:

4. Radius of Curvature, Center of Curvature and Evolutes

• Radius of Curvature $\rho$: The reciprocal of curvature:

• Osculating Circle (Circle of Curvature): The circle in the osculating plane that has the same tangent, normal, and curvature as the curve at that point. Its radius is $\rho$.
• Center of Curvature $(\alpha, \beta)$ for plane curves:

• Evolute: The locus of the centers of curvature of a given curve is called its evolute.

1. Torsion $\tau$ of a Space Curve

While curvature $\kappa$ measures the rate at which $\vec{T}$ turns away from the tangent line, torsion $\tau$ measures how sharply the space curve twists out of its osculating plane.

Because $\vec{B}(s) \cdot \vec{B}(s) = 1$, $\frac{d\vec{B}}{ds}$ is perpendicular to $\vec{B}$. Furthermore, differentiating $\vec{B} \cdot \vec{T} = 0$:

Thus, $\frac{d\vec{B}}{ds}$ is perpendicular to both $\vec{B}$ and $\vec{T}$, which means it must be parallel to $\vec{N}$. We define the scalar torsion $\tau$ by:

The negative sign is conventional so that a right-handed screw has positive torsion.

General Parametric Formula for Torsion:

Criterion for Planar Curves: A space curve is a plane curve if and only if $\tau(t) \equiv 0$ for all $t$.

2. The Complete Frenet-Serret Formulas

The rate of change of the moving frame $\{\vec{T}, \vec{N}, \vec{B}\}$ with respect to arc length $s$ is governed by the celebrated Frenet-Serret Formulas:

In matrix notation:

Notice that the coefficient matrix (the Darboux matrix) is skew-symmetric, which is a mathematical guarantee that the orthonormal nature of the basis is preserved along the entire curve.

3. Tangential and Normal Components of Acceleration

In physical kinematics, the acceleration of a particle can be decomposed uniquely into orthogonal components along the tangent and principal normal:

where:

• Tangential Acceleration: Rate of change of speed:

• Normal (Centripetal) Acceleration: Tendency to change direction:

Notice that the binormal component of acceleration is always identically zero ($a_B = 0$). Acceleration always lies entirely in the osculating plane!