Theory of curves- Space curves, Tangent, Contact of curve and surface, Osculating plane, Principal normal and Binormal, Curvature, Torsion, Serret-Frenet's formulae.

 

Osculating circle and Osculating sphere, Existence and Uniquenss theorems, Bertrand curves, Involute, Evolutes. 

 

Ruled surface, Developable surface, Tangent plane to a ruled surface. Necessary and sufficient condition that a surface ℑ=f () should represent a developable surface. 

 

Metric of a surface, First, second and third fundamental forms. Fundamental magnitudes of some important surfaces, Orthogonal trajectories. normal curvature, Meunier's theorem,.

 

Principal directions and Principal curvatures, First curvature, Mean curvature, Gaussion curvature. Umbilics. Radius of curvature of any normal section at an umbilic on z = f(x,y). Radius of curvature of a given section through any point on z = f(x,y). Lines of curvature, Principal radii, Relation between fundamental forms. Asymptotic lines.