is a Bertrand curve, where a, [xi] are constant numbers.
The spherical curve f is a circle if and only if the corresponding Bertrand curve is a circular helix.
Let [??] : S [right arrow] [R.sup.3] be an isometric immersion of a surface S in the Euclidean 3-space and [??](v) be a v-parameter curve ofthe constant slope surface [??](u, v).Then [[integral].sup.v.sub.0] [??](v)dv is a Bertrand curve, .
If there exists a spatial curve [bar.[gamma]]([bar.s]) whose principal normal direction coincides with that of original curve, then [gamma] is said to be a Bertrand curve. The pair ([gamma],[[bar.[gamma]]) is said to be a Bertrand mate.
Let [gamma] be a Bertrand curve with a[kappa] + b[tau] = 1 and [bar.[gamma]] a Bertrand mate.
Then [alpha] is a constant curvature curve, [beta] is a constant torsion curve and [gamma] is a Bertrand curve. Conversely, every Bertrand curve can be represented in this form.
If there exists another regular curve [??] such that the principal normals of c and [??] at each pair of corresponding points coincide with the line joining corresponding points then [subset] is called a Bertrand curve and the curve [??] is called a Bertrand conjugate of c.
(ii) the line joining corresponding points s, [??] of c and [??] is orthogonal to c and [??] at the points s, [??] respectively, and is along the principal normal to c or [??] at the points s, [??] whenever it is well defined, then c is called a weakened Bertrand curve and denoted by W B curve.The curve [??] is said to be a W B conjugate of c.
Classical differential geometry of the curves may be surrounded by the topics which are general helices, involute-evolute curve couples, spherical curves and Bertrand curves
. Such special curves are investigated and used in some of real world problems like mechanical design or robotics by well-known Frenet- Serret equations.
On spacelike constant slope surfaces and Bertrand curves
in Minkowski 3-space.
For instance, Bertrand curves
and Mannheim curves arise from this relationship.
Well-known partner curves are the Bertrand curves
, which are defined by the property that at the corresponding points of two space curves the principal normal vectors are common.