在 Bresenham 线算法中追踪所有 8 个八度的图示

先决条件： Bresenham 的线生成算法

在本文中，我们将了解如何使用 Bresenham’s Line Algorithm 来查找所有八度音程。

在理解算法的同时，仅考虑两点P ₀ (x ₀ , y ₀ )和P ₁ (x ₁ , y ₁ ) 的坐标使得_{x 0} < x ₁和y ₀ < y _{1 的情况}。任务是在像素的计算机屏幕上找到绘制直线 P ₀ (x ₀ , y ₀ )和P ₁ (x ₁ , y ₁ )所需的所有中间点。

下面是 Bresenham’s Line Algorithm 中使用的算法：

plotLine(x0, y0, x1, y1)
    dx = x1 - x0
    dy = y1 - y0
    D = 2*dy - dx
    DE = 2*dy
    DNE = 2(dy - dx)
    y = y0
    x = x0
    plot(x, y)
    while (x < x1)
        if D > 0
            D = D + DNE
            y = y + 1
        else
               D = D + DE
        end if
        x = x + 1
        plot(x, y)
    end while

以下是在其他八度音阶中绘制一条线应进行的更改：

• 八度音阶：无。
• Octave 2：交换 x 和 y 的角色。
• Octave 3：切换x和y的角色；使用规则(8)。
• 八度音阶：从 P1 画到 P0；使用规则(8)。
• 八度音阶：从 P1 到 P0。
• 八度音阶：从 P1 画到 P0；使用规则(2)。
• Octave 7：交换 x 和 y 的角色；使用规则(4)。
• 八度音阶：使用 y = y – 1。

下面是第一个八度的图示：

P0 = (5, 8) and  P1 = (9, 11)
dx = x1 – x0 = 9 – 5 = 4
dy = y1 – y0 = 11 – 8 = 3
D = 2*dy – dx = 2(3) – 4 = 2
DE = 2*dy = 2*3 = 6
DNE = 2(dy – dx) = 2(3 – 4) = -2

最初，绘制(x ₀ , y ₀ )即(5, 8)

• 情节(5, 8)
• D = 2 -> NE 被选中 -> plot(6, 9) -> D = 2 + DNE = 2 – 2 = 0。
• D = 0 -> E 被选中 -> plot(7, 9) -> D = 2 + DE = 0 + 6 = 6。
• D = 6 -> NE 被选中 -> plot(8, 10) -> D = 6+ DNE = 6 – 2 = 4。
• D = 4 -> NE 被选中 -> plot(9, 11) -> D = 4 + DNE = 4 – 2 = 2。
• x = x1 所以退出 while 循环。

绘制的点是 (5, 8), (6, 9), (7, 9), (8, 10), (9, 11)。

下面是第二个八度的图示：

P0 = (2, 6) and  P1 = (5, 15)

=> Perform the following steps for the third octave:

• Switch roles of x and y mean swap(x0, y0) and swap(x1, y1) and while plotting we have to plot(y, x)
• New points become P0 = (6, 2) and  P1 = (15, 5)

dx = x1 – x0 = 15 – 6 = 9
dy = y1 – y0 = 5 – 2 = 3
D = 2*dy – dx = 2(3) – 9 = 6 – 9 = -3
DE = 2*dy = 2*3 = 6
DNE = 2(dy – dx) = 2(3 – 9) = -12

最初，我们必须绘制(y ₀ , x ₀ )即(2, 6)

• 绘图(2, 6)。
• D = -3 -> E 被选中 -> plot(2, 7) -> D = -3 + DE = -3 + 6 = 3。
• D = 3 -> NE 被选中 -> plot(3, 8).-> D = 3 + DNE = 3 + -12 = -9。
• D = -9 -> E 被选中 -> plot(3, 9) -> D = -9 + DE = -9 + 6 = -3。
• D = -3 -> E 被选中 -> plot(3, 10) -> D = -3 + DE = -3 + 6 = 3。
• D = 3 -> NE 被选择 -> plot(4, 11).-> D = 3 + DNE = 3 + -12 = -9。
• D = -9 -> E 被选中 -> plot(4, 12) -> D = -9 + DE = -9 + 6 = -3。
• D = -3 -> E 被选中 -> plot(4, 13) -> D = -3 + DE = -3 + 6 = 3。
• D = 3 -> NE 被选择 -> plot(5, 14).-> D = 3 + DNE = 3 + -12 = -9。
• D = -9 -> E 被选中 -> plot(5, 15) -> D = -9 + DE = -9 + 6 = -3。
• x = x1 所以退出 while 循环。

绘制的点是 (2, 6), (2, 7), (3, 8), (3, 9), (3, 10), (4, 11), (4, 12), (4, 13) ), (5, 14), (5, 15)。

下面是第三个八度的图示：

P0 = (-2, 4) and P1 = (-6, 12)

Perform the following steps for the third octave:

• Switch roles of x and y = swap(x0, y0) and swap(x1, y1) and while plotting we have to plot(y, x)
• Use y = y – 1 = we have to use y = y – 1 instead of y = y + 1 and dy = -dy.
• New points become P0 = (4, -2) and  P1 = (12, -6)

dx = x1 – x0 = 12 – 4 = 8
dy = y1 – y0 = -6 – (-2) = -4

We have to make dy = -dy. Therefore, dy = +4

D = 2*dy – dx = 2(4) – 8 = 8 – 8 = 0
DE = 2*dy = 2*4 = 8
DNE = 2(dy – dx) = 2(4 – 8) = -8

最初，我们必须绘制 (y ₀ , x ₀ ) 即 (-2, 4)

• 绘图（-2, 4）。
• D = 0 -> E 被选中 -> plot(-2, 5) -> D = 0 + DE = 0 + 8 = 8。
• D = 8 -> NE 被选中 -> plot(-3, 6).-> D = 8 + DNE = 8 + -8 = 0。
• D = 0 -> E 被选中 -> plot(-3, 7) -> D = 0 + DE = 0 + 8 = 8。
• D = 8 -> NE 被选中 -> plot(-4, 8).-> D = 8 + DNE = 8 + -8 = 0。
• D = 0 -> E 被选中 -> plot(-4, 9) -> D = 0 + DE = 0 + 8 = 8。
• D = 8 -> NE 被选中 -> plot(-5, 10).-> D = 8 + DNE = 8 + -8 = 0。
• D = 0 -> E 被选中 -> plot(-5, 11) -> D = 0 + DE = 0 + 8 = 8。
• D = 8 -> NE 被选中 -> plot(-6, 12).-> D = 8 + DNE = 8 + -8 = 0。
• x = x1 所以退出 while 循环。

绘制的点是 (-2, 4), (-2, 5), (-3, 6), (-3, 7), (-4, 8), (-4, 9), (-5, 10), (-5, 11), (-6, 12)。

下面是第四个八度的图示：

P0 = (-2, 1) and  P1 = (-6, 3)

Perform the following steps for the fourth octave:

• Draw from P1 to P0 = swap(P0, P1).
• Use y = y – 1 = we have to use y = y – 1 instead of y = y + 1 and dy = -dy.
• New points become P0 = (-6, 3) and  P1 = (-2, 1) */

dx = x1 – x0 = -2 – (-6) = 4
dy = y1 – y0 = 1 – 3 = -2

We have to make dy = -dy. Therefore, dy = +2

D = 2*dy – dx = 2(2) – 4 = 4 – 4 = 0
DE = 2*dy = 2*2 = 4
DNE = 2(dy – dx) = 2(2 – 4) = -4

最初，我们必须绘制 (x0, y0) 即 (-6, 3)

• 绘图（-6, 3）。
• D = 0 -> E 被选中 -> plot(-5, 3) -> D = 0 + DE = 0 + 4 = 4。
• D = 4 -> NE 被选中 -> plot(-4, 2).-> D = 4 + DNE = 4 + -4 = 0。
• D = 0 -> E 被选中 -> plot(-3, 2) -> D = 0 + DE = 0 + 4 = 4。
• D = 4 -> NE 被选中 -> plot(-2, 1).-> D = 4 + DNE = 4 + -4 = 0。
• x = x1 所以退出 while 循环。

绘制的点是 (-6, 3), (-5, 3), (-4, 2), (-3, 2), (-2, 1)。

下面是第五个八度的图示：

P0 = (-3, -1) and  P1 = (-9, -3)

Perform the following steps for the fifth octave:

• Draw from P1 to P0 = swap(P0, P1).
• New points become P0 = (-9, -3) and  P1 = (-3, -1)

dx = x1 – x0 = -3 – (-9) = 6
dy = y1 – y0 = -1 – (-3) = 2
D = 2*dy – dx = 2(2) – 6 = 4 – 6 = -2
DE = 2*dy = 2*2 = 4
DNE = 2(dy – dx) = 2(2 – 6) = -8

最初，我们必须绘制 (x0, y0) 即 (-9, -3)

• 绘图（-9，-3）。
• D = -2 -> E 被选中 -> plot(-8, -3) -> D = -2 + DE = -2 + 4 = 2。
• D = 2 -> NE 被选择 -> plot(-7, -2).-> D = 2 + DNE = 2 + -8 = -6。
• D = -6 -> E 被选中 -> plot(-6, -2) -> D = -6 + DE = -6 + 4 = -2。
• D = -2 -> E 被选中 -> plot(-5, -2) -> D = -2 + DE = -2 + 4 = 2。
• D = 2 -> NE 被选中 -> plot(-4, -1).-> D = 2 + DNE = 2 + -8 = -6。
• D = -6 -> E 被选中 -> plot(-3, -1) -> D = -6 + DE = -6 + 4 = -2。
• x = x1 所以退出 while 循环。

绘制的点是 (-9, -3), (-8, -3), (-7, -2), (-6, -2), (-5, -2), (-4, -1 ), (-3, -1)。

下面是第六个八度的图示：

P0 = (-1, -2) and  P1 = (-3, -6)

Perform the following steps for the sixth octave:

• Draw from P1 to P0 = swap(P0, P1).
• Switch roles of x and y = swap(x0, y0) and swap(x1, y1) and while plotting we have to plot(y, x).
• New points become P0 = (-6, -3) and  P1 = (-2, -1)

dx = x1 – x0 = -2 – (-6) = 4
dy = y1 – y0 = -1 – (-3) = 2
D = 2*dy – dx = 2(2) – 4 = 4 – 4 = 0
DE = 2*dy = 2*2 = 4
DNE = 2(dy – dx) = 2(2 – 4) = -4

最初，我们必须绘制 (y0, x0) 即 (-3, -6)

• 绘图（-3，-6）。
• D = 0 -> E 被选中 -> plot(-3, -5) -> D = 0 + DE = 0 + 4 = 4。
• D = 4 -> NE 被选中 -> plot(-2, -4).-> D = 4 + DNE = 4 + -4 = 0。
• D = 0 -> E 被选中 -> plot(-2, -3) -> D = 0 + DE = 0 + 4 = 4。
• D = 4 -> NE 被选中 -> plot(-1, -2).-> D = 4 + DNE = 4 + -4 = 0。
• x = x1 所以退出 while 循环。

绘制的点是 (-3, -6), (-3, -5), (-2, -4), (-2, -3), (-1, -2)。

下面是第七个八度的图示：

P0 = (2, -5) and  P1 = (4, -10)

Perform the following steps for the seventh octave:

• Switch roles of x and y = swap(x0, y0) and swap(x1, y1) and while plotting we have to plot(y, x).
• Draw from P1 to P0 = swap(P0, P1).
• Use y = y – 1 = we have to use y = y – 1 instead of y = y + 1 and dy = -dy.
• New points become P0 = (-10, 4) and  P1 = (-5, 2)

dx = x1 – x0 = -5 -(-10) = 5
dy = y1 – y0 = 2 – 4 = -2

We have to make dy = -dy. Therefore, dy = +2

D = 2*dy – dx = 2(2) – 5 = 4 – 5 = -1
DE = 2*dy = 2(2) = 4
DNE = 2(dy – dx) = 2(2 – 5) = -6

最初，我们必须绘制 (y0, x0) 即 (4, -10)

• 绘图（4，-10）。
• D = -1 -> E 被选中 -> plot(4, -9) -> D = -1 + DE = -1 + 4 = 3。
• D = 3 -> NE 被选择 -> plot(3, -8).-> D = 3 + DNE = 3 + -6 = -3。
• D = -3 -> E 被选中 -> plot(3, -7) -> D = -3 + DE = -3 + 4 = 1。
• D = 1 -> NE 被选中 -> plot(2, -6).-> D = 1 + DNE = 1 + -6 = -5。
• D = -5 -> E 被选中 -> plot(2, -5) -> D = -5 + DE = -5 + 4 = -1。
• x = x1 所以退出 while 循环。

绘制的点是 (-4, 10), (-4, 9), (-3, 8), (-3, 7), (-2, 6), (-2, 5)。

下面是第八个八度的图示：

P0 = (3, -2) and  P1 = (6, -4)

Perform the following steps for the eighth octave:

• Use y = y – 1 = we have to use y = y – 1 instead of y = y + 1 and dy = -dy.

dx = x1 – x0 = 6 – 3 = 3
dy = y1 – y0 = -4 – (-2) = -2

We have to make dy = -dy. Therefore, dy = +2

D = 2*dy – dx = 2(2) – 3 = 4 – 3 = 1
DE = 2*dy = 2*2 = 4
DNE = 2(dy – dx) = 2(2 – 3) = -2

最初，我们必须绘制 (x ₀ , y0) 即 (3, -2)

• 绘图（3，-2）。
• D = 1 -> NE 被选中 -> plot(4, -3) -> D = 1 + DNE = 1 + (-2) = -1。
• D = -1 -> E 被选中 -> plot(5, -3).-> D = -1 + DE = -1 + 4 = 3。
• D = 3 -> NE 被选中 -> plot(6, -4) -> D = 3 + DNE = 3 + -2 = 1。
• x = x1 所以退出 while 循环。

绘制的点是 (3, -2), (4, -3), (5, -3), (6, -4)。

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