Zerlegung der Darboux-Drehung in Zwei Ebene Drehungen

Adapted Frame and Darboux Type Vector of Spacelike Framed Curves

“…Minkowski 3-space, denoted ℝ 1 3 , is a threedimensional vector space equipped with a metric tensor <, > that provides a measure of interval or distance between events in the spacetime of special relativity. The Lorentzian metric <, > which has symmetric, bilinear and non-degenerate properties is defined as follows for vectors 𝑢 = (𝑢 1 , 𝑢 2 , 𝑢 3 ) and 𝑣 = (𝑣 1 , 𝑣 2 , 𝑣 3 ) < 𝑢, 𝑣 >= −𝑢 1 𝑣 1 + 𝑢 2 𝑣 2 + 𝑢 3 𝑣 3 .…”

“…A vector 𝑤 ∈ ℝ 1 3 is called spacelike vector if < 𝑤, 𝑤 >> 0 or 𝑤 = 0, timelike vector if < 𝑤, 𝑤 >< 0 or a lightlike vector if < 𝑤, 𝑤 >= 0 and 𝑤 ≠ 0. A curve 𝛾: 𝐼 ⊂ ℝ → ℝ 1 3 is called a spacelike, timelike, or lightlike curve at 𝑡 ∈ 𝐼 ⊂ ℝ if its velocity vector 𝛾′(𝑡) is a spacelike, timelike or lightlike vector, respectively (see [8,10]). Since we study the geometry of the curve, we can assign an orthonormal frame to any point on a smooth regular curve.…”

“…A spacelike curve 𝛾 in ℝ 1 3 can contain singular points, that is, points 𝑠₀ ∈ 𝐼 such that 𝛾 ′ (𝑠 0 ) = 0. Then the Frenet frame of the curve 𝛾 cannot be constructed.…”

“…To solve this kind of problem, the authors in [6] extend the notion of framed curve introduced in [4] to the Minkowski 3-space. They define spacelike framed curves in ℝ 1 3 , where a spacelike curve with a collection of moving frame fields along its length is referred to as a spacelike framed curve [6,7].…”

“…The angular velocity vector of a space curve's Frenet frame is known as the Darboux vector. In [1] and [3], an iterative method was proposed for the relevant Darboux axis rotation motions for Frenet curves, resulting in a series of Darboux vectors. Yücesan and Tükel [14] offer a more comprehensive perspective on the Darboux type vector of the Frenet-type frame for framed base curves that might contain singular points and derive a series of Darboux vectors.…”