Reference no: EM132262691

Question - Orthogonal Families of Curves

a. Let E, F, G be the coefficients of the first fundamental form of a regular surface S in the parametrization x: U ⊂ R² → S. Let φ(u, v) = const. and ψ(u, v) = const. be two families of regular curves on x(U) ⊂ S. Prove that these two families are orthogonal (i.e., whenever two curves of distinct families meet, their tangent lines are orthogonal) if and only if

Eφ_vψ_v - F(φ_uψ_v + φ_vψ_u) + Gφ_uψ_u = 0.

b. Apply part a to show that on the coordinate neighborhood x(U) of the helicoid of Example 3 the two families of regular curves

v cos u = const., v ≠ 0,

(v² + a²)sin² u = const., v ≠ 0, u ≠ π,

are orthogonal.