In this section, we recall C-Bézier and H-Bézier basis functions proposed by Wang et al. A free shape parameter is introduced in these basis functions to portray polynomial and circular curves of high degree precisely.

### Definition 1

The C-Bézier basis function \(\{C_{i}^{n}(t) \}_{i=0}^{n}\) for the space \(T=\operatorname{span} \{1,t, \ldots ,t^{n-2}, \sin t, \cos t \}\) of degrees *n* is defined as follows

$$ \begin{aligned} &C_{0}^{n}(t)=1- \int _{0}^{t}\delta _{0}^{n-1}C_{0}^{n-1}(s)\,ds, \\ &C_{i}^{n}(t)= \int _{0}^{t}\delta _{i-1}^{n-1}C_{i-1}^{n-1}(s)\,ds- \int _{0}^{t}\delta _{i}^{n-1}C_{i}^{n-1}(s)\,ds, \quad i=1,2,\ldots ,n-1, \\ &C_{n}^{n}(t)= \int _{0}^{t}\delta _{n-1}^{n-1}C_{n-1}^{n-1}(s)\,ds, \end{aligned} $$

(1)

where \(n\geq 2\), \(t\in [0,\alpha ]\), \(\alpha \in (0,\pi ]\), \(C_{0}^{1}(t)=\frac{\sin(\alpha -t)}{\sin\alpha}\), \(C_{1}^{1}(t)=\frac{\sin t}{\sin\alpha}\), \(\delta _{i}^{n}=(\int _{0}^{\alpha }C_{i}^{n}(t)\,dt)^{-1}\), *α* is referred as shape parameter [31].

While \(n=2\), C-Bézier basis function is expressed as follows.

$$ \begin{aligned} &C_{0}^{2}(t)= \frac{1-\cos(\alpha -t)}{1-\cos\alpha}, \\ &C_{1}^{2}(t)=-\frac{1-\cos t+\cos\alpha -\cos(\alpha -t)}{1-\cos\alpha}, \\ &C_{2}^{2}(t)=\frac{1-\cos t}{1-\cos\alpha}, \end{aligned} $$

(2)

where \(t\in [0,\alpha ]\), \(\alpha \in (0,\pi ]\). Figure 1(a) shows the quadratic C-Bézier basis function at \(\alpha =\frac{\pi}{6}\). Figure 1(b) presents the quadratic C-Bézier basis function at \(\alpha =\frac{3\pi}{4}\).

C-Bézier basis functions are a generalization of classical Bernstein basis functions that pose a lot of common properties, such as affine invariance, endpoints interpolation, convex hull property. From the graph above, we can see that the C-Bézier basis functions have symmetry written by the mathematical formula as

$$ C_{i}^{n}(t)=C_{n-i}^{n}(\alpha -t)\quad \text{for } t\in [0,\alpha ], i=0,1, \ldots ,n.$$

### Definition 2

A C-Bézier curve \(p(t)\) is defined by

$$ p(t)=\sum_{i=0}^{n}P_{i}C_{i}^{n}(t),\quad t \in [0,\alpha ],$$

here, \(P_{i}\) (\(i=0,1,\ldots ,n\)) is called control point, \(C_{i}^{n}(t)\) is the C-Bézier basis function for the space \(T=\operatorname{span} \{1,t,\ldots ,t^{n-2},\sin t,\cos t \}\), *α* is the shape parameter.

Figure 2(a) provides the picture of quadratic curve with control points \((0,0)\), \((1,5)\), \((3,2)\) and \(\alpha =\frac{\pi}{4}\). Figure 2(b) displays a cubic C-Bézier curve with \(\alpha =\pi \), it’s control points are \((0,0)\), \((2,5)\), \((3,1)\), \((4,4)\). Figure 2(c) presents the figure of a quadratic C-Bézier curve with control points \((0,8)\), \((2,4)\), \((4,2)\), \((6,3)\), \((8,8)\) and \(\alpha =2\).

### Definition 3

The H-Bézier basis function \(\{H_{i}^{n}(t) \}_{i=0}^{n}\) for the space \(T=\operatorname{span} \{1,t, \ldots , t^{n-2}, \sinh t, \cosh t \}\) of *n* degrees is defined by

$$ \begin{aligned} &H_{0}^{n}(t)=1- \int _{0}^{t}\delta _{0}^{n-1}H_{0}^{n-1}(s)\,ds, \\ &H_{i}^{n}(t)= \int _{0}^{t}\delta _{i-1}^{n-1}H_{i-1}^{n-1}(s)\,ds- \int _{0}^{t}\delta _{i}^{n-1}H_{i}^{n-1}(s)\,ds,\quad i=1,2,\ldots ,n-1, \\ &H_{n}^{n}(t)= \int _{0}^{t}\delta _{n-1}^{n-1}H_{n-1}^{n-1}(s)\,ds, \end{aligned} $$

(3)

in which \(n\geq 2\), \(t\in [0,\alpha ]\), \(\alpha \in (0,+\infty )\), \(H_{0}^{1}(t)=\frac{\sinh(\alpha -t)}{\sinh\alpha}\), \(H_{1}^{1}(t)=\frac{\sinh t}{\sinh\alpha}\), \(\delta _{i}^{n}=(\int _{0}^{\alpha}H_{i}^{n}(t)\,dt)^{-1}\), *α* is called shape parameter [30].

While \(n=2\), the H-Bézier basis function is expressed by Eq. (4).

$$ \begin{aligned} &H_{0}^{2}(t)= \frac{1-\cosh(\alpha -t)}{1-\cosh\alpha}, \\ &H_{1}^{2}(t)=-\frac{1-\cosh t+\cosh\alpha}{1-\cosh\alpha}+ \frac{\cosh(\alpha -t)}{1-\cosh\alpha}, \\ &H_{2}^{2}(t) =\frac{1-\cosh t}{1-\cosh\alpha}, \end{aligned} $$

(4)

here, \(t\in [0,\alpha ]\), \(\alpha \in (0,+\infty )\). The quadratic H-Bézier basis functions at \(\alpha =\pi \) and \(\alpha =8\) are shown in Fig. 3. Basis functions at \(\alpha =\pi \) and \(\alpha =8\) are also shown in Fig. 3.

### Definition 4

An H-Bézier curve \(p(t)\) is defined by

$$ p(t)=\sum_{i=0}^{n}P_{i}H_{i}^{n}(t),\quad t \in [0,\alpha ].$$

Similarly to C-Bézier curve, \(P_{i}\) (\(i=0,1,\ldots ,n\)) is control point, \(H_{i}^{n}(t)\) is the H-Bézier basis function for the space \(T=\operatorname{span} \{\sinh t,\cosh t,t^{k-3},t^{k-4},\ldots ,t,1 \}\), *α* is the shape parameter.

As shown in Fig. 4, the graph on the left demonstrates a quadratic H-Bézier curve with control points \((0,5)\), \((2,2)\), \((4,7)\) and \(\alpha =2\). The image of a cubic curve with \(\alpha =3 \) is in the middle; its control points are \((0,6)\), \((1,2)\), \((3,4)\), \((4,1)\). A quadratic H-Bézier curve with control points \((0,0)\), \((2,4)\), \((4,7)\), \((6,4)\), \((8,1)\) and \(\alpha =1 \) is provided on the right.

The H-Bézier basis functions have many beneficial properties in common with Bernstein basis functions. Compared to classical Bernstein basis, C-Bézier and H-Bézier basis functions have a free variable allowing adjustment of the local or global shape and geometrical properties of the curves. In addition, even though shape parameter may change, the position and tangent direction of the curve’s endpoints do not alter [35]. Furthermore, in the splicing of multiple C-Bézier curves, the shape parameter will not transform the original continuity between curves [36].