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Torsion — Definition, Formula & Examples

Torsion is a scalar quantity that measures how much a space curve twists out of the plane defined by its tangent and normal vectors (the osculating plane). Positive torsion indicates a right-handed twist, while negative torsion indicates a left-handed twist.

For a smooth space curve r(s)\mathbf{r}(s) parametrized by arc length with nonzero curvature, the torsion τ\tau is defined by the Frenet–Serret relation dBds=−τ N\frac{d\mathbf{B}}{ds} = -\tau\,\mathbf{N}, where B\mathbf{B} is the unit binormal vector and N\mathbf{N} is the principal unit normal vector.

Key Formula

τ=(r′×r′′)⋅r′′′∥r′×r′′∥2\tau = \frac{\left(\mathbf{r}' \times \mathbf{r}''\right) \cdot \mathbf{r}'''}{\left\|\mathbf{r}' \times \mathbf{r}''\right\|^2}
Where:
  • τ\tau = Torsion of the curve
  • r′\mathbf{r}' = First derivative of the position vector with respect to parameter t
  • r′′\mathbf{r}'' = Second derivative of the position vector
  • r′′′\mathbf{r}''' = Third derivative of the position vector

How It Works

While curvature tells you how sharply a curve bends within a plane, torsion tells you how that plane itself is rotating as you move along the curve. A curve with zero torsion everywhere is a planar curve. To compute torsion from an arbitrary parametrization r(t)\mathbf{r}(t), you use the cross product r′×r′′\mathbf{r}'\times\mathbf{r}'' and the triple scalar product (r′×r′′)⋅r′′′\left(\mathbf{r}'\times\mathbf{r}''\right)\cdot\mathbf{r}'''. Together, curvature and torsion completely determine the shape of a space curve up to rigid motion (translation and rotation), a result known as the fundamental theorem of space curves.

Worked Example

Problem: Find the torsion of the circular helix r(t) = (2cos t, 2sin t, 3t).
Compute derivatives: Find the first three derivatives of r(t).
r′=(−2sin⁡t,  2cos⁡t,  3),r′′=(−2cos⁡t,  −2sin⁡t,  0),r′′′=(2sin⁡t,  −2cos⁡t,  0)\mathbf{r}' = (-2\sin t,\; 2\cos t,\; 3), \quad \mathbf{r}'' = (-2\cos t,\; -2\sin t,\; 0), \quad \mathbf{r}''' = (2\sin t,\; -2\cos t,\; 0)
Cross product: Compute r' × r''.
r′×r′′=(6sin⁡t,  −6cos⁡t,  4)\mathbf{r}' \times \mathbf{r}'' = (6\sin t,\; -6\cos t,\; 4)
Numerator (triple scalar product): Dot the cross product with r'''.
(6sin⁡t)(2sin⁡t)+(−6cos⁡t)(−2cos⁡t)+(4)(0)=12sin⁡2t+12cos⁡2t=12(6\sin t)(2\sin t) + (-6\cos t)(-2\cos t) + (4)(0) = 12\sin^2 t + 12\cos^2 t = 12
Denominator: Find the squared magnitude of the cross product.
∥r′×r′′∥2=36sin⁡2t+36cos⁡2t+16=52\|\mathbf{r}' \times \mathbf{r}''\|^2 = 36\sin^2 t + 36\cos^2 t + 16 = 52
Torsion: Divide to get τ.
τ=1252=313\tau = \frac{12}{52} = \frac{3}{13}
Answer: The torsion of the helix is τ = 3/13, a positive constant, confirming the helix twists uniformly in a right-handed sense.

Why It Matters

Torsion is essential in differential geometry and in engineering applications such as designing roads, roller coasters, and DNA modeling, where understanding how a curve twists through three-dimensional space directly affects structural and physical behavior.

Common Mistakes

Mistake: Confusing torsion with curvature, or assuming that a curve with large curvature must also have large torsion.
Correction: Curvature measures bending within the osculating plane; torsion measures rotation of that plane. A circle has constant curvature but zero torsion because it is planar.