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, .
Oral: Bertrand Partner D-Curves in Euclidean 3-space E3, arXiv:1003.2044v3 [math.DG], 2010.
As introduced (as ansatz) in  and as shall be derived formally in the two parts of this paper, the hyper-surface (or proper Euclidean 3-space) [SIGMA]' along the horizontal is underlied by an isotropic one-dimensional proper intrinsic space denoted by [phi][rho]' (that has no unique orientation in the Euclidean 3-space [SIGMA]').
Inclusion of the proper time dimension ct' along the vertical, normal to the horizontal hyper-surface (or horizontal Euclidean 3-space) [SIGMA]' in Fig.
In this paper, we would like to contribute the solution of the above question, by studying this question for tubes or tubular surfaces in Euclidean 3-space [E.sup.3] and Minkowski 3-space [E.sup.3.sub.1].
Thus, M is an open part of a circular cylinder in Euclidean 3-space. We have the following theorem and corollary:
Here using the standard coordinate system of Euclidean 3-space [E.sup.3], a surface r(u, v) in [E.sup.3] will be written as r(u, v) = (x(u, v), y(u, v), z(u, v)).The surface which can be written as (1.2) is usually called translation surface in Euclidean 3-space [E.sup.3] (, ).
In this note we define affine translation surfaces z(x, y) = f (x)+ g(y + ax) in Euclidean 3-space [E.sup.3] and get the following main result.
Equation (1) states that the one-dimensional space [[rho].sup.0]' along the vertical projects zero component (or nothing) into the Euclidean 3-space [SIGMA]' (as a hyper-surface) along the horizontal.
where [phi][rho]' is without the superscript "0" label because it lies in (or underneath) our Euclidean 3-space [SIGMA]' (without superscript "0" label) along the horizontal.
Point [A.sup.*] in the negative Euclidean 3-space -[[SIGMA].sup.*] of the negative universe is the symmetry-partner to point A in the positive Euclidean 3-space [SIGMA] of the positive universe.
Every particle or object with a three-dimensional inertial mass m in the Euclidean 3-space [SIGMA] has its one-dimensional intrinsic mass to be denoted by [phi]m underlying it in the one-dimensional intrinsic space [phi][rho].