Formula derivation and code implementation of Bezier curve

This article only briefly describes the formula derivation of the Bezier curve, and gives a code implementation. Before reading this article, please make sure you have some background knowledge of Bezier curves. Relevant knowledge can be learned through the following courses: MOOC-Computer Graphics-China Agricultural University-Zhao Ming or watch the port version of Station B.

Algorithm principle

Given $n + 1$ control point $P_i(i=0,1,\cdots,n)$ , structure polynomial function
$\mathcal{P}(u)=\sum_{i= 0}^{n}P_i\cdot BEZ_{i,n}(u), \quad 0\leqslant u\leqslant 1 \thinspace, \tag{1}$
Approach curve. Among them, $BEZ_{i,n}(u)$ called $n$ 次Bernstein基函数，且有
$BEZ_{i,n}(u)=C_n^i u^i(1-u)^{n-i}, \quad \left(\begin{array}{l} i=0,1,\cdots,n \\ u\in[0,1] \end{array}\right) \thinspace。 \tag{2}$

For formula (1), the function $\mathcal{P}(u)$ converts the real number interval $[0, 1]$ is mapped to the point set on the real number field, and the points in the point set form the approximate curve sought. $BEZ_{i,n}(u)$ can be regarded as the control point $P_i$ in The weight under $u .$ Because $BEZ_{i,n}(u)$ 具有权性：
$\sum_{i=0}^{n} BEZ_{i,n}(u) \equiv 1, \quad \forall u\in(0,1) \thinspace。\tag{3}$

我们定义：
$\beta_{i}^{(r)} = \begin{cases} P_i, & r = 0 \\ (1-u)\beta_{i}^{(r-1)} + u\beta_{i+1}^{(r-1)}, & r = 1,2,\cdots,n \end{cases} \thinspace， \tag{4}$
i = 0 , 1 , ⋯ , n $ri=0,1,\cdots,nr$ 。

It can be proved (see Wikipedia De Casteljau algorithm for the proof process ):
$\mathcal{P}(u) = \sum_{i=0}^{n} \beta_{i}^{(0)}\cdot BEZ_{i,n}(u) = \beta_{0 }^{(n)} \thinspace. \tag{5}$

So the algorithm logic is as follows:

In the interval of real numbers $[0, 1]$ over a large number of samples. For each sample $u$ , computation $\mathcal{P}(u)$ 。
According to formula (5), we can calculate $\beta_{0}^{(n)}$ To calculate $\mathcal{P}(u)$ 。
$\beta_{0}^{(n)}$ It can be calculated recursively by formula (4).

Code

def draw_curve(p_list):
	"""
    :param p_list: (list of list of int: [[x0, y0], [x1, y1], [x2, y2], ...]) 曲线的控制点坐标列表
    :return: (list of list of int: [[x_0, y_0], [x_1, y_1], [x_2, y_2], ...]) 绘制结果的像素点坐标列表
    """
    result = []
    n = len(p_list) - 1
    freq = 1000
    for u in range(freq + 1):
        u /= freq
        tmp = [[p[0], p[1]] for p in p_list]
        for r in range(1, n + 1):
            tmp = [
                [(1 - u) * tmp[i][0] + u * tmp[i + 1][0],
                 (1 - u) * tmp[i][1] + u * tmp[i + 1][1]
                 ] for i in range(n - r + 1)
            ]
        result.append([round(tmp[0][0]), round(tmp[0][1])])
    return result