Website owner:  James Miller

Differential Geometry

Curves in the plane, derivative of arc length, curvature, radius of curvature, circle of curvature, evolute

Space curves, tangent vector, principal normal, binormal, curvature, torsion, Frenet-Serret formulas

Lines and planes in space

Vector functions of a vector variable, directional derivatives

Surfaces, surface representation, simple surface elements, curvilinear coordinates, surface normals, surface curves, first and second fundamental quadratic forms, osculating paraboloid, surface area

Surface curvature: normal, total and mean curvature, Euler's theorem, Meusnier's theorem, umbilical point, Rodrigues' Formula, lines of curvature

Curvature of a curve on a surface

Dupin indicatrix, asymptotic directions and lines

Christoffel symbols of the first and second kind

Gauss-Weingarten equations, Gauss-Codazzi equations, Fundamental Theorem of Surfaces

Geodesics, geodesic curvature, geodesic parallels, geodesic coordinates, geodesic torsion, Gauss-Bonnet theorem

Surface maps, conformal maps, isothermic surfaces, area-preserving maps

Isometric mapping, intrinsic property

Envelopes, characteristics, tangent surface of a space curve, ruled surfaces, developable surfaces