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We describe a robust method for constructing a tubular surface surrounding a spacelike curve with a spacelike binormal in Minkowski 3-Space. Our method is designed to eliminate undesirable twists and wrinkles in the tubular surface’s skin at points where the curve experiences high torsion. In our construction the tubular surface’s twist is bounded by the spacelike curve’s curvature and is independent of the spacelike curve’s torsion. 1. Preliminaries Let IR = {(x1, x2, x3) | x1, x2, x3 ∈ IR} be a 3-dimensional vector space, and let x = (x1, x2, x3) and y = (y1, y2, y3) be two vectors in IR . The Lorentz scalar product of x and y is defined by 〈x, y〉L = x1y1 − x2y2 + x3y3, IE1 = ( IR, 〈x, y〉L ) is called 3-dimensional Lorentzian space, Minkowski 3Space or 3dimensional Semi-Euclidean space. The vector x in IE1 is called a spacelike vector, null vector or a timelike vector if 〈x, x〉L > 0 or x = 0, 〈x, x〉L = 0 or 〈x, x〉L < 0, respectively. For x ∈ IE 3 1, the norm of the vector x defined by ‖x‖L = √ |〈x, x〉L|, and x is called a unit vector if ‖x‖L = 1. For any x, y ∈ IE 3 1 , Lorentzian vectoral product of x and y is defined by x ∧L y = (x2y3 − x3y2, x1y3 − x3y1, x1y2 − x2y1) . Similarly, an arbitrary curve α = α(s) in IE1 is locally spacelike, timelike or null (lightlike), if all of its velocity vectors α′(s) = T (s) are respectively spacelike, timelike or null, for each s ∈ I ⊂ IR. The vectors v, w ∈ IE1 are orthogonal if and only if 〈v, w〉L = 0. The Lorentzian sphere of center m = (m1, m2, m3) and radius r ∈ R+ in the space IE1 is defined by [3] S 1 = { a = (a1, a2, a3) ∈ IE 3 1 | 〈a − m, a − m〉L = r 2 }

An alternative moving frame for tubular surface around the spacelike curve with a spacelike binormal in Minkowski 3-space