In this paper, pedal-like curves are defined resulting from the orthogonal projection of a fixed point on the alternative frame vectors of a given regular curve. For each pedal curve, the Frenet vectors, the curvature and the torsion functions are found to provide the common relations among the main curve and its pedal curves. Then, Smarandache curves are defined by using the alternative frame vectors of each pedal curve as position vectors. The relations of the Frenet apparatus are also established for the pedal curves and their corresponding Smarandache curves. Finally, the expressions of the alternative frame apparatus of each Smarandache curves are given in terms of the alternative frame elements of the pedal curves. Thus, a set of new symmetric curves are introduced that contribute to the vast curve family.

The purpose of this paper is to construct relations and characterizations between the pedal curves and surfaces and to find components of the vector of α(t) by the means of the pedal on the surface M. Also, the formula of pedal curves of a curve α is generalized in n‐dimensional Euclidean space Eⁿ. Some special results are obtained within the scope of pedal curves and given with obtained characterization of the pedal curve in n‐dimensional space Eⁿ.

In this paper, the involute and evolute of the curve is studied in type of the curve-surface pair at first time. In additional whenβ is considered evolute and involute of the curveα, involute and evolute curve-surface pairs (called as strip) and strip of the curveαis shown as(α,M) are given with depending on the constant angleφthat is between in and in Euclidean 3-Space E^3.

The spherical helix and the strip are respectively proved firstly by Scofield and Sabuncuoglu and Hacısalihoglu. In this paper helix strip on sphere is investigated by using characteristics of spherical helix and strip. Firstly using strip after helixand finally spherical helix. So spherical helix strips are obtained. Furhermore Joachimsthal Theorem and Terquem Theorem are investigated when the strip and helix strips which lie on the sphere and given a characterization about spherical helix strips.

Willmore defined embedded surfaces on $f:S\to E^3$, which is the embedding of S into Euclidean 3-space. He investigated the Euclidean metric of $E^3$, inducing a Riemannian structure on $f\left(S\right)$. The expression analogous to the left-hand member of the curvature K is replaced by the mean curvature $H^2$ on $f\left(S\right).$ Our aim is to observe the Gaussian and mean curvatures of curve–surface pairs using embedded surfaces in different curve–surface pairs and to define some developable operations on their curve–surface pairs. We also investigate the embedded surfaces using the Willmore method. We first recall the Darboux curve–surface and derive the new characterizations. This curve–surface pair is called the osculating Darboux curve–surface if its position vector always lies in the osculating Darboux plane spanned by a Darboux frame. Thus, we observed an osculating Darboux curve–surface pair. We also obtained the $\stackrel{\sim }{D}$-scroll of the curve–surface pair and involute $\stackrel{\sim }{D}$-scroll of the curve–surface pair with some differential geometric elements and found ${\stackrel{\sim }{D}}_{(\alpha ,M)}\left(s\right)$ and ${\stackrel{\sim }{D^{\ast }}}_{(\alpha ,M)}\left(s\right)$-scrolls of the curve–surface pair $(\alpha ,M)$.