Curve construction based on five trigonometric blending functions (Q695052)

For \(\alpha,\, \beta \geq 2\) and \(t\in [0,\pi/2]\), the authors consider the trigonometric blending functions \[ \begin{cases} & T_0(t)=(1-\sin t)^{\alpha},\\ & T_1(t) = \alpha\, \sin t\,(1 - \sin t)^{\alpha -1},\\ & T_3(t) = \beta\, \cos t\, (1- \cos t)^{\beta -1},\\ & T_4(t) = (1 - \cos t)^{\beta}\end{cases} \] and \[ T_2(t) = \sum_{j\neq 2} T_j(t) \] which form a nonnegative partition of unity. For given points \(P_j \in {\mathbb R}^d\) \((j=0,\ldots,4)\), \(d=2\) or \(d=3\), and \(t\in [0,\pi/2]\), a trigonometric Bézier curve with the shape parameters \(\alpha,\, \beta \geq 2\) is defined by \[ T(t;\,\alpha,\beta) = \sum_{j=0}^4 P_j\, T_j(t)\,. \] For convenient points \(P_j\in {\mathbb R}^2\) \((j=0,\ldots,4)\), \(T(t;\,2,2)\) is an elliptic arc or a parabolic arc. The authors discuss the smoothness of composite trigonometric Bézier curves, too. Finally, trigonometric blending functions are used to construct trigonometric B-spline curves.