Differential Geometry glossary

Clear, one-line definitions of the Differential Geometry terms used across the OgbonLab textbooks. Each entry links to the interactive sections where the idea is taught.

angular defect

For a vertex of a polyhedron, 2π minus the sum of the face angles meeting there; total defect equals 2π·χ (Descartes).

curvature

A measure of how sharply a curve bends; κ = ‖T'(s)‖, the rate at which the unit tangent T turns per unit arc length.

See: Curvature of Plane Curves, Curvature and the Three Geometries

gauss-bonnet theorem

For a compact surface M, ∫∫_M K dA = 2π·χ(M), linking total Gaussian curvature to the Euler characteristic.

See: The Gauss-Bonnet Theorem

normal section

For a surface at a point p, the curve cut by a plane containing the surface normal; its curvature is the normal curvature.

osculating circle

At a point on a smooth curve, the unique circle tangent to the curve there with the same curvature; radius 1/κ.

osculating plane

For a space curve at a point, the plane spanned by the unit tangent and principal normal vectors; best fits the curve locally.

principal curvatures

At a point on a surface, the maximum and minimum normal curvatures κ₁ and κ₂; their product is the Gaussian curvature.

radius of curvature

At a point on a curve, the reciprocal R = 1/κ of the curvature; the radius of the osculating circle.

signed curvature

For a plane curve, a signed scalar whose magnitude is κ and whose sign indicates left or right turning relative to orientation.

torsion

For a space curve, the scalar τ measuring how sharply it twists out of its osculating plane; τ = 0 iff the curve is planar.

See: Curvature and Torsion of Space Curves