Bresenham 的 3-D 线图算法

给定两个 3-D 坐标，我们需要找到连接它们的线上的点。所有点都有整数坐标。

例子：

输入：(-1, 1, 1), (5, 3, -1) 输出：(-1, 1, 1), (0, 1, 1), (1, 2, 0), (2, 2) , 0), (3, 2, 0), (4, 3, -1), (5, 3, -1) 输入：(-7, 0, -3), (2, -5, -1) 输出：(-7, 0, -3), (-6, -1, -3), (-5, -1) , -3), (-4, -2, -2), (-3, -2, -2), (-2, -3, -2), (-1, -3, -2), ( 0, -4, -1), (1, -4, -1), (2, -5, -1)

Bresenham 算法是高效的，因为它避免了浮点算术运算。在二维线图的情况下，我们使用一个变量来存储斜率误差，即从实际几何线绘制的线的斜率误差。一旦斜率误差超过允许值，我们就修改数字以消除误差。

要绘制的线的驱动轴是线沿其行进最远的轴，即轴坐标的差异最大。因此，坐标值沿驱动轴线性增加 1，斜率误差变量用于确定另一个轴的坐标值的变化。

在 2-D 线的情况下，我们使用一个斜率误差变量，但在 3-D 线的情况下，我们需要两个 ( $py, pz$ ) 对于每个非驱动轴。如果当前点是$P_{k}$ (x, y, z)且驱动轴为正 X 轴，则下一个点$P_{k+1}$可能

• (x+1, y, z)
• (x+1, y+1, z)
• (x+1, y, z+1)
• (x+1, y+1, z+1)

斜率误差变量的值根据以下等式确定：-
$py_{k+1} = py_{k} + 2dy - 2dx(y_{k+1} - y_{k})\\ pz_{k+1} = pz_{k} + 2dz - 2dx(z_{k+1} - z_{k})$

斜率误差变量的初始值由以下等式给出：-
$py_{0} = 2dy - dx\\ pz_{0} = 2dz - dx$
这里$dx, dy, dz$表示两个端点沿 X、Y、Z 轴的坐标差。

算法：-

1. 输入两个端点并将初始点存储为$(x_{0}, y_{0}, z_{0})$
2. 阴谋$(x_{0}, y_{0}, z_{0})$
3. 计算常数$dx, dy, dz$并通过比较确定驱动轴
   的绝对值$dx, dy, dz$
   如果绝对（ $dx$ ) 为最大值，则 X 轴为驱动轴
   如果绝对（ $dy$ ) 为最大值，则 Y 轴为驱动轴
   如果绝对（ $dz$ ) 为最大值，则 Z 轴为驱动轴
4. 假设X轴是驱动轴，那么
   $py_{0} = 2dy - dx\\ pz_{0} = 2dz - dx$
5. 在每一个$x_{k}$沿线，从 k = 0 开始，检查以下条件
   并确定下一点：-
   • 如果$py_{k} < 0$和$pz_{k} < 0$ ， 然后
     阴谋$(x_{k}+1, y_{k}, z_{k})$和
     放$py_{k+1}=py_{k}+2dy, pz_{k+1}=pz_{k}+2dz$
   • 否则如果$py_{k} > 0$和$pz_{k} < 0$ ， 然后
     阴谋$(x_{k}+1, y_{k}+1, z_{k})$和
     放$py_{k+1}=py_{k}+2dy-2dx, pz_{k+1}=pz_{k}+2dz$
   • 否则如果$py_{k} 0$ ， 然后
     阴谋$(x_{k}+1, y_{k}, z_{k}+1)$和
     放$py_{k+1}=py_{k}+2dy, pz_{k+1}=pz_{k}+2dz-2dx$
   • 否则
     阴谋$(x_{k}+1, y_{k}+1, z_{k}+1)$和
     放$py_{k+1}=py_{k}+2dy-2dx, pz_{k+1}=pz_{k}+2dz-2dx$ >
6. 重复步骤 5 $dx-1$次

Python3

# Python3 code for generating points on a 3-D line 
# using Bresenham's Algorithm
  
def Bresenham3D(x1, y1, z1, x2, y2, z2):
    ListOfPoints = []
    ListOfPoints.append((x1, y1, z1))
    dx = abs(x2 - x1)
    dy = abs(y2 - y1)
    dz = abs(z2 - z1)
    if (x2 > x1):
        xs = 1
    else:
        xs = -1
    if (y2 > y1):
        ys = 1
    else:
        ys = -1
    if (z2 > z1):
        zs = 1
    else:
        zs = -1
  
    # Driving axis is X-axis"
    if (dx >= dy and dx >= dz):        
        p1 = 2 * dy - dx
        p2 = 2 * dz - dx
        while (x1 != x2):
            x1 += xs
            if (p1 >= 0):
                y1 += ys
                p1 -= 2 * dx
            if (p2 >= 0):
                z1 += zs
                p2 -= 2 * dx
            p1 += 2 * dy
            p2 += 2 * dz
            ListOfPoints.append((x1, y1, z1))
  
    # Driving axis is Y-axis"
    elif (dy >= dx and dy >= dz):       
        p1 = 2 * dx - dy
        p2 = 2 * dz - dy
        while (y1 != y2):
            y1 += ys
            if (p1 >= 0):
                x1 += xs
                p1 -= 2 * dy
            if (p2 >= 0):
                z1 += zs
                p2 -= 2 * dy
            p1 += 2 * dx
            p2 += 2 * dz
            ListOfPoints.append((x1, y1, z1))
  
    # Driving axis is Z-axis"
    else:        
        p1 = 2 * dy - dz
        p2 = 2 * dx - dz
        while (z1 != z2):
            z1 += zs
            if (p1 >= 0):
                y1 += ys
                p1 -= 2 * dz
            if (p2 >= 0):
                x1 += xs
                p2 -= 2 * dz
            p1 += 2 * dy
            p2 += 2 * dx
            ListOfPoints.append((x1, y1, z1))
    return ListOfPoints
  
  
def main():
    (x1, y1, z1) = (-1, 1, 1)
    (x2, y2, z2) = (5, 3, -1)
    ListOfPoints = Bresenham3D(x1, y1, z1, x2, y2, z2)
    print(ListOfPoints)
  
main()

输出：

[(-1, 1, 1), (0, 1, 1), (1, 2, 0), (2, 2, 0), (3, 2, 0), (4, 3, -1), (5, 3, -1)]

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