Differential geometry

Start here

• ConceptDifferentiable curve

  A differentiable curve is a parametrised curve whose coordinate functions have continuous derivatives up to some order $k \ge 1$. It is the basic object of the differential geometry of curves. Tangent vectors, arc length, curvature and torsion are all defined for differentiable curves of suitable order.

• ConceptArc length

  The arc length of a curve is its length, measured along the curve. It is the supremum of the lengths of inscribed polygonal paths, and an integral of the speed for smooth curves. The node arc length gives the precise definition.

• ConceptCurvature of a curve

  The curvature of a curve measures how fast its direction turns per unit of length travelled. For a unit-speed curve it is the length of the acceleration vector. A straight line has curvature $0$ and a circle of radius $r$ has curvature $1/r$. Curvature, together with torsion, determines a space curve up to a rigid motion.

• ConceptTorsion of a curve

  The torsion of a space curve measures how fast the curve twists out of its osculating plane. This is the plane that best fits the curve at a point. A plane curve has torsion $0$ everywhere. With curvature, torsion appears in the Frenet–Serret formulas and determines a space curve up to a rigid motion.

• TheoremFrenet–Serret formulas

  Let $I \subseteq \R$ be an interval and let $\gamma \colon I \to \R^3$ be a curve of class $C^3$ parametrised by arc length $s$. Assume that its curvature $\kappa(s)$ is positive for all $s \in I$. Let $T = \gamma'$, $N = T' / \kappa$ and $B = T \times N$, and let $\tau$ be the torsion of $\gamma$. Then $(T(s), N(s), B(s))$ is a positively oriented orthonormal basis of $\R^3$ for each $s$, and $$T' = \kappa N, \qquad N' = -\kappa T + \tau B, \qquad B' = -\tau N.$$

• TheoremFundamental theorem of space curves

  Let $I \subseteq \R$ be an open interval, and let $\kappa \colon I \to \R$ and $\tau \colon I \to \R$ be smooth functions with $\kappa(s) > 0$ for all $s \in I$. Then there is a smooth curve $\gamma \colon I \to \R^3$ parametrised by arc length whose curvature is $\kappa$ and whose torsion is $\tau$. If $\tilde\gamma \colon I \to \R^3$ is another such curve, then $\tilde\gamma = \rho \circ \gamma$ for a unique orientation-preserving rigid motion $\rho$ of $\R^3$. Here a rigid motion is a map $x \mapsto Rx + b$ with $R$ a rotation matrix and $b \in \R^3$.