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A pedal curve (a contrapedal curve) of a regular plane curve is the locus of the feet of the perpendiculars from a point to the tangents (normals) to the curve. These curves can be parametrized by using the Frenet frame of the curve. Yet provided that the curve has some singular points, the Frenet frame at these singular points is not well-defined. Thus, we cannot use the Frenet frame to examine pedal or contrapedal curves. In this paper, pedal and contrapedal curves of plane curves, which have singular points, are investigated. By using the Legendrian Frenet frame along a front, the pedal and contrapedal curves of a front are introduced and properties of these curves are given. Then, the condition for a pedal (and a contrapedal) curve of a front to be a frontal is obtained. Furthermore, by considering the definitions of the evolute, the involute, and the offset of a front, some relationships are given. Finally, some illustrated examples are presented.

A generalized helicoid and a rotational surface have an isometric relationby Bour's theorem. It is that "A generalized helicoid is isometric to arotational surface. Hence, helices on the helicoid can be transformed toparallel circles on the rotational surface under the isometrictransformation".In this study, we give a conformal relation between a generalized helicoid(with lightlike profile curve) and a spiral surface (with lightlike profilecurve). In this sitiutation, we can say that helices on the helicoid can betransformed to spirals on the spiral surface under the conformaltransformation. Also, some related examples and their figures are given.2000 Mathematics Subject Classification. 53A05, 53C10S, Bir genelleştirilmiş helikoid ve bir dönel yüzey arasında Bour teoremi ile izometrik bir ilişkisi vardır. Bu "bir genelleştirilmiş helikoid bir dönel yüzeye izometrik. Dolayısıyla, helikoid üzerindeki helisler izometrik dönüşüm altında dönel yüzey üzerindeki paralel çemberlere karşılık gelir." olmasıdır. Bu çalışmada, bir genelleştirilmiş helikoid (lightlike profil eğrisi ile) ve bir spiral yüzey (lightlike profil eğrisi ile) arasında bir konformal ilişki verilmiştir. Bu durumda, bir helikoid üzerindeki helislerin konformal dönüşüm altında spiral yüzey üzerindeki spirallere karşılık geldiğini söyleyebiliriz. Ayrıca, bu konuyla ilgili bazı örnekler ve bu örneklerin şekilleri verilmiştir.

In this paper, we introduce an inclined curves according to parallel transport frame. Also, we define a vector field called Darboux vector field of an inclined curve in and we give a new characterization such as: "\alpha: I \subset R \rightarrow E^4 is an inclined curve \Leftrightarrow k_1 \int k_1ds + k_2 \int \k_2 +k_3ds = 0" where k_1, k_2, K_3 are the principal curvature functions according to parallel transport frame of the curve and we give the similar characterizations such as "\alpha : I \subset R \rightarrow E^3 is a generalized helix \Leftrightarrow k_1 \int k_1ds + k_2 \int k_2ds = 0" where k_1, k_2 are the principal curvature functions according to Bishop frame of the curve \alpha. Moreover, we illustrate some examples and draw their figures with Mathematica Programme.