Differential Geometry of Curves and Surfaces

By Thomas F. Banchoff, Stephen Lovett. 2010

1,010 citations recorded by the source corpus.

Plane Curves: Local Properties Parametrizations Position, Velocity, and Acceleration Curvature Osculating Circles, Evolutes, and Involutes Natural Equations Plane Curves: Global Properties Basic Properties Rotation Index Isoperimetric Inequality Curvature, Convexity, and the Four-Vertex Theorem Curves in Space: Local Properties Definitions, Examples, and Differentiation Curvature, Torsion, and the Frenet Frame Osculating Plane and Osculating Sphere Natural Equations Curves in Space: Global Properties Basic Properties Indicatrices and Total Curvature Knots and Links Regular Surfaces Parametrized Surfaces Tangent Planes and Regular Surfaces Change of Coordinates The Tangent Space and the Normal Vector Orientable Surfaces The First and Second Fundamental Forms The First Fundamental Form Map Projections (Optional) The Gauss Map The Second Fundamental Form Normal and Principal Curvatures Gaussian and Mean Curvature Developable Surfaces and Minimal Surfaces The Fundamental Equations of Surfaces Gauss's Equations and the Christoffel Symbols Codazzi Equations and the Theorema Egregium The Fundamental Theorem of Surface Theory The Gauss-Bonnet Theorem and Geometry of Geodesics Curvatures and Torsion Gauss-Bonnet Theorem, Local Form Gauss-Bonnet Theorem, Global Form Geodesics Geodesic Coordinates Applications to Plane, Spherical and Elliptic Geometry Hyperbolic Geometry Curves and Surfaces in n-Dimensional Euclidean Space Curves in n-Dimensional Euclidean Space Surfaces in Rn Appendix: Tensor Notation

Subfields: Computational Mechanics

View Paper: https://www.semanticscholar.org/paper/4a4c1c7d80086cafd334e2066389288b637713b9

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In Computational Mechanics

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4. Bo Zhou
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Citations in the corpus

• Directional Tubular Surfaces (2015, background): “There are a number of different adapted frames along a space curve, such as the parallel transport frame [1,12] and the Frenet frame [5].”
• Three-dimensional sphericity, roundness and fractal dimension of sand particles (2018, methodology): “, 2014; Hanaor et al., 2016). Sphericity and roundness have traditionally been used to classify and quantify the morphological properties of soils in engineering and geology (Mehring & McBride, 2007; Santamarina & Cho, 2014). According to Wadell (1935), sphericity is quantified by comparing the surf”
• Multifunctional Mechanical Metamaterials Based on Triply Periodic Minimal Surface Lattices (2019, background): “In general, a minimal surface is a surface that has a mean curvature equal to zero at every point, H 1⁄4 0, where H 1⁄4 ðk1 þ k2Þ=2 and k1 and k2 are the principal curvatures in two mutually orthogonal planes.([16]) The divergence of the unit normal vector is zero everywhere and the Gaussian curvatu”
• Full-Body Musculoskeletal Model for Muscle-Driven Simulation of Human Gait (2016, background): “Because there is an analytic solution for how a muscle path wraps over a cylindrical surface [31], we expected this would improve simulation speed.”
• Implicit Geometric Regularization for Learning Shapes (2020, background): “…this analysis locally noting that locally an MLP is approximately linear and the deviation from planarity is quantified locally by the curvature of the surface, e.g., the two principle surface curvatures σ 1 , σ 2 that represent the reciprocal radii of two osculating circles (Do Carmo, 2016).”
• Report Shape and Dynamics of Tip-Growing Cells (2009, neutral): “To study the interplay between growth and mechanics in shaping a walled cell, we examine the particularly simple geometry of tip-growing cells [1, 3, 15, 16], which elongate via the assembly and expansion of cell wall in the apical region of the cell.”
• Intrinsic curvature of curves and surfaces and a Gauss–Bonnet theorem in the Heisenberg group (2016, background): “In the classical theory of differential geometry of smooth space curves in R3, there is a famous rigidity theorem (see for instance [23]) stating that every curve is characterized by its curvature and its torsion, up to rigid motions.”
• CAT – A Computational Anatomy Toolbox for the Analysis of Structural MRI Data (2023, neutral): “Cortical Folding : CAT provides distinct cortical folding measures, which are derived from the geometry of the central surface: ‘Gyrification’ is calculated via the absolute mean curvature (Carmo, 2016) of the central surface.”
• Mannheim partner curves in 3-space (2008, neutral): “It is well known that a twisted curve in E3 is a generalized helix if and only if the ratio τ/κ is a nonzero constant (see [1]).”