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1 Basic Concepts and Previous Studies
1.1 Introduction
1.2 The three-dimensional Galilean space G
1.2.1 Curves in Galilean space G
1.2.2 Bishop frames
1.3 Natural geometry of ruled surfaces in G
1.3.1 Darboux frame of a curve lying on the ruled surface
of type I
1.3.2 Darboux frame of a curve lying on the ruled surface
of type III
1.4 Helices in G
1.5 Bertrand curves in G
1.6 Geometry ofthe pseudo¡ªGalilean space G
1.6.1 Curves in pseudo-Galilean space
1.6.2 Bishop frames in Gi
1.7 Natural geometry of ruled surfaces in G
1.7.1 Darboux frame of a curve lying on a ruled surface in
G
1.8 Helices in G
1.9 Normal and rectifying curves in Gj
2 Spherical Indicatrices of Helices in Galilean Spaces
2.1 Introduction
2.2 Spherical images of special curves in Galilean space
2.2.1 A unit speed curve
2.2.2 Spherical curves of the position vector of an arbitrary
CUrVe
2.2.3 Bishop spherical images of an arbitrary curve
2.3 Example
2.4 Spherical curves in pseudo-Galilean space
2.4.1 Spherical indicatrices of an arbitrary curve
2.5 spherical images with Bishop frame
2.6 Spherical images with Bishop frame of a circular helix
2.7 Spherical images with Bishop frame of Salkowski curve
2.8 Spherical images with Bishop frame of Anti¡ªSalkowski curve
2.9 Examples
3 Smarandache Curves of Helices in the Galilean 3-Space
3.1 Geonmtric prelinfinaries
3.2 Special Smarandac,he(turves in Galilean geometry
3.2.1 Smarandache curves of a unit speed curve
3.3 Relations among spherical indicatrices of SOUlC Smarandache curves
3.3.1 Smarandache curves of all arbitrary curve with respect to standard frame
3.4 Special Smarandache curves according to Darboux frame in G
3.5 Exmnples
3.6 Smaremdache curves of special curves iIl pseudo-Galilean geolnetry
3.6.1 Smarandache curves of aIl arbitrary curve
3.6.2 Special Slnarandache curves according to Darboux fraum
3.7 Exainples
4 Bertrand Curves in the Galilean and Pseudo¡ªGalilean Spaces
4.1 Introductioll
4.2 Bertrand partner curves accoroding to Freimt fraiue
4.3 Bertrand curves according to Darboux franm in G3
4.4 Exainples
4.5 Bertrand curves ill pseudo-Oalilean geometry
4.6 Bertrand curve~according to Darboux flame in G3/1
4.7 Examples
5 Normal, Osculating and Rectifying Curves in Galilean Spaces
5.1 Introductiou
5.2 Bishop frame of the second type in G3
5.3 Associated curves according to Bishop fralne in G.
5.4 Associated curves in the pseudo-Galilean space G3/1
Bibliography
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