√$_{3}$ 2 cosh ( s ) + √$_{3}$ 2 t sinh ( s ) + cosh ( − 1 $_{2}$s ) cosh ( s ) − 1 2 sinh ( − 1 $_{2}$s ) sinh ( s ) ) where 0 ≤ s ≤ 2 π, − 2 ≤ t ≤ 2. Fig. 3.2. P$_{2}$ ( s, t ) as a member of the surface pencil and its line of curvature Let P = P ( s, t ) be a parametric timelike surface and r = r ( s ) be a spacelike curve with timelike binormal. The surface is defined by the given curve as L$_{1}$ ≤ s ≤ L$_{2}$, T$_{1}$ ≤ t ≤ T$_{2}$, where { T ( s ) , N ( s ) , B ( s ) } is the Frenet frame associated with the curve r ( s ) . Since the curve r ( s ) is a parametric curve on the surface P ( s, t ), there exists 12 P ( s, t ) = r ( s ) + ( u ( s, t ) , v ( s, t ) , w ( s, t ))    T ( s ) N ( s ) B ( s )   $^{,}$ (4.1) 4 Timelike Surface Pencil with a Common Line of Curvature