Parametric representation of a surface pencil with a common asymptotic curve

Surface pencils in Euclidean 4-space 𝔼4

“…However, Kasap et al generalized the marching-scale functions given in [13] and gave a sufficient condition for a given curve to be a geodesic on a surface [7]. Recently, Bayram et al extend the method given in [13] to derive the necessary and sufficient condition for a given curve to be both isoparametric and asymptotic on a parametric surface [2]. Also, Ergün et al considered surface pencil with a common line of curvature in Minkowski 3-space [4].…”

Hypersurface Family with a Common Isoasymptotic Curve

“…al. [6] tackled the problem of finding a surface pencil from a given asymptotic curve. However, while differential geometry of a parametric surface in R 3 can be found in textbooks such as in Struik [21], Willmore [24], Stoker [20], do Carmo [7], differential geometry of a parametric surface in R n can be found in textbooks such as in the contemporary literature on Geometric Modeling [9], [13].…”

Rotation-minimizing osculating frames

“…In terms of these quantities, the geodesics, line of curvatures, and asymptotic lines on a smooth surface may be characterized [25,32] as loci along which κ g ≡ 0, τ g ≡ 0, and κ n ≡ 0, respectively. From Definition 1, we obtain the following alternative succinct characterizations, which are useful in the construction of surface patches bounded by geodesics [19,29,31]; lines of curvature [4,26,27]; and asymptotic lines [2]. …”