Title
On Ruled Surfaces in threedimensional Minkowski Space,Used
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In a Minkowski three dimensional space we define a semiinnerproduct based on the socalled cosineMinkowski function. We also construct an orthogonal 3D frame in Birkhoff sense, which is canonically adapted to ruled surfaces: beginning with the generator direction we complete this frame using the strictly convex and centrally symmetric unit ball B, which is described either by supporting function or vector representation. Based on the leftorthogonality defined by ball B, the striction curve of a ruled surface in a Minkowski 3space can be declared in analogy to the Euclidean case. We define the new vector called Deformation vector which helps us to find the FrenetSerret formulae of the ruled surface in the Minkowski three dimension spaces. In these formulae we insert the Mcurvatures and MTorsions with respect to the Minkowski frame. We also can define a covariant differentiation in a Minkowski 3space, with this can declare geometric Mparallelity of the vector field of the generator of a skew ruled surface along its Minkowski striction curve. Using the second fundamental form the relation between Euclidean and Minkowski normal vectors is given.
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We provide a 2-year limited warranty, from the date of purchase for all our products.
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This warranty strictly does not cover damages that arose from negligence, misuse, wear and tear, or not in accordance with product instructions (dropping the product, etc.).
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