Math 3355 Geometry of Curves & Surfaces

Topics

• Parametrisations
• position, velocity, and acceleration
• curvature
• evolutes
• and involutes
• and natural equations
• Basic properties of plane curves
• rotation index
• isoperimetic inequality
• curvature, convexity, and the four-vertex theorem
• Definitions, examples, and differentiation of curves in space
• torsion, and the Frenet frame
• osculating plane and osculating sphere
• natural equations
• Basic properties of curves in space, indicatrices and total curvature
• knots and links
• coordinates
• The tangent space and the normal vector, orientable surfaces
• The first fundamental form, the Gauss map, the second fundamental
• form
• Normal and principal curvatures, gaussian and mean curvatures, ruled
• surfaces and minimal surfaces
• Tensor notation, Gauss’ equations, and the Christoffel symbols
• Codazzi equations and the Theorema Egregium, the fundamental
• theorem of surface theory