Differential Geometry of Curves and Surfaces
Summary
Differential geometry of curves and surfaces examines how smooth lines and two-dimensional manifolds bend and twist within ambient spaces. Beginning with the pioneering work on Frenet–Serret formulae for space curves and Gauss’s investigation of intrinsic surface curvature, the field has grown to encompass global theorems, such as the Gauss–Bonnet result, and sophisticated techniques like mean curvature flow and Ricci flow. Curvature and torsion quantify local bending of curves, while the first and second fundamental forms capture metric and extrinsic bending of surfaces. Beyond pure mathematics, these concepts underpin modern physics through general relativity, influence material science via thin‐shell theory, and drive computer graphics and architectural design by providing algorithms for smooth shape modelling and optimisation.
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Differential Geometry of Curves and Surfaces publication trend
The graph below shows the total number of articles in differential geometry of curves and surfaces across all publications each year (not limited to Nature Index journals).
Technical terms
Frenet frame: Orthogonal triad of tangent, normal and binormal vectors along a space curve, encoding its local curvature and torsion.
Gaussian curvature: Intrinsic measure of surface curvature given by the product of the two principal curvatures at a point.
Mean curvature: Average of the principal curvatures at a surface point, indicating bending in ambient space.
Gauss map: Mapping that assigns to each surface point the unit normal vector on the sphere, capturing orientation data.
Ruled surface: Surface generated by sweeping a straight line (the generator) along a guiding curve (the directrix).
Developable surface: Special ruled surface with zero Gaussian curvature that can be unfolded onto a plane without distortion.
Singularity: Point where a curve or surface fails to be regular, often exhibiting cusps, self‐intersections or degenerate tangent behaviour.
References
- Geometry of Solutions of the Geometric Curve Flows in Space. Electronic Journal of Applied Mathematics (2023).
- The developable surfaces with pointwise 1-type Gauss map of Frenet type framed base curves in Euclidean 3-space. AIMS Mathematics (2022).
- Singularities for Timelike Developable Surfaces in Minkowski 3-Space. Symmetry (2023).
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