Advanced Modeling 2

where x(t), y(t) and z(t) are differentiable functions in t. The vector ċ is called tangent vector. A curve point c(t0), t0 ∈ I is called regular if ċ(t0) 6= o. A parametrization is called regular, if ċ(t) 6= o for all t ∈ I. Any differentable change of the parameter τ = τ(t) does not change the curve. Moreover, if τ̇ 6= 0 in I then d(t) = c(τ(t)) is also a regular parametrization. A curve which admits a regular parametrization is called regular. A parametric surface is given by s : s(u, v) =  x(u, v) y(u, v) z(u, v)  , (u, v) ∈ [a, b]× [c, d] = I × J ⊂ R