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The curvature of a space curve measures turning of its unit tangent vector; the torsion of a curve measures twisting of its osculating plane. For a regular curve $r(t)$ in Euclidean space ${\mathbb{R}}^3$,The first formula requires two derivatives and $r^{\prime }\ne 0$. The second requires three derivatives and $r^{\prime }\times r^{\prime \prime }\ne 0$: torsion of a curve is undefined by this formula at zero curvature of a space curve.

With arc length $s$ and Frenet frame $(T,N,B)$, the formulas become $\kappa =\Vert dT/ds\Vert$ and $\tau =-⟨dB/ds,N⟩$. A circle of radius $a>0$ has $\kappa =1/a$ and $\tau =0$. The circular helix $r(t)=(a\cos t,a\sin t,bt)$ has $\kappa =a/(a^2+b^2)$ and $\tau =b/(a^2+b^2)$. These data, expressed in arc length, determine the curve up to a proper Euclidean motion of Euclidean three-space by the fundamental theorem of regular space curves.^[ref]

Let $\alpha (s)$ be a regular curve parametrized by arc length on the unit sphere, with Frenet frame $(T,N,B)$ and convention $B^{\prime }=-\tau N$. Differentiating $⟨\alpha ,\alpha ⟩=1$ gives $⟨\alpha ,T⟩=0$ and $⟨\alpha ,N⟩=-1/\kappa$. ThusDifferentiating the second inner product identity using the Frenet-Serret formulas gives $\tau \beta ={\kappa }^{\prime }/{\kappa }^2$. Wherever $\tau \ne 0$,For the outward surface unit normal vector $\alpha$, the signed geodesic curvature isBoth plus signs follow from the stated torsion of a curve and surface orientation conventions. Also $1={\kappa }^{-2}+{\beta }^2$, hence ${\kappa }^2=1+{\kappa }_g^2$. The identities involving $\beta$ remain valid when $\tau =0$; the divided formulas need not. A non-equatorial latitude is a planar circle with zero torsion of a curve but nonzero geodesic curvature.

Mark Powell studies the topology of manifolds, especially in dimensions three and four, together with homological algebra. His Glasgow profile verifies this research specialization.^[ref]

His Introduction to Differential Geometry by Mark Powell connects the local geometry of curves and surfaces with global invariants. Curvature and torsion of a curve describe how a curve bends in space, whereas the Gauss-Bonnet theorem turns integrated surface curvature and boundary terms into an Euler characteristic. These examples show how local differential measurements can constrain a whole geometric object.^[ref]