资源下载地址：https://download.csdn.net/download/sheziqiong/86768948
资源下载地址：https://download.csdn.net/download/sheziqiong/86768948

基本图形生成算法

直线段

基础算法

计算斜率和截距，通过y = kx + b的直线表达式计算每一个x对应的y值

'''基础算法'''
def drawLine_Basic(grid, start, end):k = (end.y-start.y)/(end.x-start.x)b = start.y - k * start.xfor xi in range(start.x, end.x):    # 栅格的性质yi = k * xi + bdrawPixel(xi, int(yi+0.5), 1, grid)     # y坐标要进行近似

数值微分算法(DDA)

采用**“增量”**的思想
- 当|k|<=1时，x每增加1，y增加k
- 当|k|>1时，y每增加1，x增加1/k
证明: (这里只考虑|k|<=1当情况)

由xi+1=xi+1x_{i+1} = x_{i} + 1xi+1=xi+1

yi+1=k∗xi+1+b=k∗(xi+1)+b=k∗xi+b+k=yi+ky_{i+1} = k*x_{i+1} + b = k*(x_{i}+1) + b = k*x_{i} + b + k = y_{i} + kyi+1=k∗xi+1+b=k∗(xi+1)+b=k∗xi+b+k=yi+k

'''数值微分算法（DDA）'''
def drawLine_DDA(grid, start, end):k = (end.y - start.y) / (end.x - start.x)xi, yi = start.x, start.yif(abs(k<=1)):for xi in range(start.x, end.x):drawPixel(xi, int(yi+0.5), 1, grid)yi += kelse:for yi in range(start.y, end.y):drawPixel(int(xi+0.5), yi, 1, grid)xi += 1/k

如果不对k进行分类讨论

不对k进行分类讨论 ![在这里插入图片描述](https://img-blog.csdnimg.cn/d1bbad90a0874dc3aaa6a9f8e584cfbe.png) 对k进行分类讨论 ------

中点画线法

设直线方程为：ax + by + c =0
- a = y0 - y1
- b = x1 - x0
- c = x0y1 - x1y0
**考核点：**(xp+1, yp+0.5)
判别式：Δ=F(xp+1,yp+0.5)=a∗(xp+1)+b∗(yp+0.5)+c\Delta = F(x_p+1, y_p+0.5) = a*(x_p+1) + b*(y_p+0.5) + cΔ=F(xp+1,yp+0.5)=a∗(xp+1)+b∗(yp+0.5)+c
- 如果Δ<0\Delta<0Δ<0 => Q点在M下方 => 选p2 (x+1, y+1)
- else，选p1 (x+1, y)

'''中点画线法(k<=1)'''
def drwaLine_MidPoint(grid, start, end):a, b, c = start.y-end.y, end.x-start.x, start.x*end.y-end.x*start.yxp, yp = start.x, start.yfor xp in range(start.x, end.x):drawPixel(xp, yp, 1, grid)delta = a*(xp+1) + b*(yp+0.5) + c   # 考核点(xp+1, yp+0.5)if delta<0:yp += 1else:# yp += 0pass

在中点画线法中添加增量的思想

若取p1，增量为a
若取p2，增量为a+b
初值：d0 = a + 0.5*b
由于只用d的符号来判断，可以用2d代替d，摆脱浮点数

'''中点画线法 with DDA'''
def drawLine_MidPoint_with_DDA(grid, start, end):a, b = start.y-end.y, end.x-start.xd = a + (b<<2)      # 用2d代替d， 摆脱小数d1, d2 = a<<2, (a+b)<<2xp, yp = start.x, start.yfor xp in range(start.x, end.x):drawPixel(xp, yp, 1, grid)if d<0:yp += 1d += d2else:d += d1

Bresenham画线法

由误差项符号决定下一个像素选正右方还是右上方
判别式：ε=yi+1−yi,r−0.5\varepsilon = y_{i+1} - y_{i,r} - 0.5ε=yi+1−yi,r−0.5
- ε>0\varepsilon > 0ε>0, 取右上 (x+1, y+1)
- else，取正右 (x+1, y)
引入增量思想：
- ε>0\varepsilon > 0ε>0，增量为 k-1
- else，增量为 k
- 初始值：-0.5

'''Bresenham画线法(k<=1)'''
def drawLine_Bresenham(grid, start, end):k = (end.y - start.y) / (end.x - start.x)x, y = start.x, start.ye = -0.5for x in range(start.x, end.x):drawPixel(x, y, 1, grid)if e > 0:e += k - 1y += 1else:e += k# y += 0

去点浮

用ε′=2∗ε∗dx\varepsilon' = 2 * \varepsilon * dxε′=2∗ε∗dx代替ε\varepsilonε
去掉k的计算
引入增量思想：
- ε>0\varepsilon > 0ε>0，增量为 2(dy - dx)
- else，增量为 2dy
- 初始值：-dx

'''Bresenham画线法(去点浮)(k<=1)'''
def drawLine_Bresenham_nonreal(grid, start, end):dx, dy = (end.x - start.x), (end.y - start.y)x, y = start.x, start.ye = -dxfor x in range(start.x, end.x):drawPixel(x, y, 1, grid)if e > 0:e += (dy - dx) << 2y += 1else:e += (dy) << 2# y += 0

圆弧

暴力算法

y2=R2−x2y^2 = \sqrt{R^2 - x^2}y2=R2−x2

中点画圆法

只需画1/8圆（第一象限 y>x 部分）
判别式：F(x,y)=x2+y2−R2F(x,y) = x^2 + y^2 - R^2F(x,y)=x2+y2−R2
- F>0，取正右方 (x+1, y)
- else，取右下方 (x+1, y-1)
增量思想：
- d<0, 增量为 2*x + 3
- else, 增量为 2*(x-y) + 5
- 初始值：1.25 - R
**去点浮：**用e = d - 0.25代替 d
- 初始值：e = 1-R
- 循环条件：d < 0 <=> e < 0.25 <= e始终为整数 => e < 0

'''中点画圆法(DDA)'''
def drawArc_MidPoint_with_DDA(grid, R):d = 1 - Rx, y = 0, Rwhile x < y:drawPixel_symmetry8(x, y, 1, grid)if d < 0:x += 1d += 2*x + 3else:x += 1y -= 1d += ((x-y) << 1) + 5

对增量本身再次使用增量思想

x递增1，d递增Δx=2\Delta x = 2Δx=2
y递减1，d递增Δy=2\Delta\ y = 2Δ y=2
初始值：
- Δx=3\Delta x = 3Δx=3
- Δy=−2r+2\Delta\ y = -2r + 2Δ y=−2r+2

'''中点画圆法(DDA)(去点浮)'''
def drawArc_MidPoint_with_DDA_nonreal(grid, R):d = 1 - Rdeltax, deltay = 3, 2 - (R << 1)x, y = 0, Rwhile x < y:drawPixel_symmetry8(x, y, 1, grid)if d < 0:x += 1d += deltaxdeltax += 2else:x += 1y -= 1d += (deltax + deltay)deltax += 2deltay += 2

Bresenham画圆法

选取距离真正的圆曲线近的点进行扩展

如果|OP1| > |OP2|，则选P2

<=> x12+y12−R2>R2−x22−y22\sqrt{x_1^2 + y_1^2 - R^2} > \sqrt{R^2 - x_2^2 - y_2^2}x12+y12−R2>R2−x22−y22

<=> x12+y12+x22+y22−2R2>0x_1^2 + y_1^2 + x_2^2 + y_2^2 - 2R^2 > 0x12+y12+x22+y22−2R2>0
当前像素的下一个扩展节点：正右方、右下方、正下方

取H
在H、D中选更近的
取D
在V、D中选更近的
取V

令ΔH=∣OH∣\Delta H = |OH|ΔH=∣OH∣, ΔD=∣OD∣\Delta D = |OD|ΔD=∣OD∣, ΔV=∣OV∣\Delta V = |OV|ΔV=∣OV∣

令δHD=∣ΔH∣−∣ΔD∣=∣OH∣−∣OD∣\delta HD = |\Delta H| - |\Delta D| = |OH| - |OD|δHD=∣ΔH∣−∣ΔD∣=∣OH∣−∣OD∣, δDV=∣ΔD∣−∣ΔV∣\delta DV = |\Delta D| - |\Delta V|δDV=∣ΔD∣−∣ΔV∣
1. δHD=2ΔD+2y−1\delta HD = 2\Delta D + 2y - 1δHD=2ΔD+2y−1
2. δDV=2ΔD−2x−1\delta DV = 2\Delta D - 2x - 1δDV=2ΔD−2x−1
  - ΔD>0\Delta D > 0ΔD>0，若δDV<=0\delta DV <= 0δDV<=0，取D；else取V
  - ΔD<0\Delta D < 0ΔD<0，δHD<=0\delta HD <= 0δHD<=0，取H；else，取D
  - ΔD=0\Delta D = 0ΔD=0，取D
用增量法简化ΔD\Delta DΔD，下一个像素取：
- H，增量取2x + 1
- D，增量取2x - 2y + 2
- V，增量取-2y + 1
- 初始值：-2r + 2

'''Bresenham画圆法'''
def drawArc_Bresenham(grid, R):delta = (1 - R) << 1x, y = 0, Rwhile y >= 0:drawPixel_symmetry4(x, y, 1, grid)if delta < 0:delta1 = ((delta + y) << 1) - 1if delta1 <= 0:direction = 1else:direction = 2elif delta > 0:delta2 = ((delta - x) << 1) - 1if delta2 <= 0:direction = 2else:direction = 3else:direction = 2if direction == 1:      # 前进到 正右x += 1delta += (x << 1) + 1elif direction == 2:    # 前进到 右下x += 1y -= 1delta += ((x - y) << 1) + 2else:                   # 前进到 正下y -= 1delta += 1 - (y << 1)

正负法

圆方程：F(x,y)=x2+y2−R2F(x,y) = x^2 + y^2 - R^2F(x,y)=x2+y2−R2
设P(xi,yi)P(x_i, y_i)P(xi,yi)
- P在圆内 -> D(xi,yi)<=0D(x_i,y_i) <= 0D(xi,yi)<=0 -> 向右
- P在圆外 -> D(xi,yi)>0D(x_i,y_i) > 0D(xi,yi)>0 -> 向下
引入增量思想：
- 当F<=0，增量为2x + 1
- else，增量为-2y + 1
- 初始值：0

'''正负法'''
def drawArc_PositiveNegative(grid, R):F = 0x, y = 0, Rwhile x <= y:drawPixel_symmetry8(x, y, 1, grid)print(F)if F <= 0:F += (x << 1) + 1x += 1else:F += 1 - (y << 1)y -= 1

圆内接正多边形逼近法

limn−>∞(n边正内接多边形)=圆lim_{n->\infin}(n边正内接多边形) = 圆limn−>∞(n边正内接多边形)=圆
- 工程上，n−>2πRn -> 2\pi Rn−>2πR就可以了，两个点的距离达到1pixel之后再小也没用了
圆的参数方程:
- xi=Rcosθix_i = Rcos\theta_ixi=Rcosθi
- yi=Rsinθiy_i = Rsin\theta_iyi=Rsinθi
引入增量思想：
(xi+1yi+1)=(cosα−sinαsinαcosα)(xiyi)=(cosαxi−sinαyisinαxi+cosαyi)(\begin{matrix}x_{i+1} \\ y_{i+1} \end{matrix}) = (\begin{matrix}cos\alpha & -sin\alpha \\ sin\alpha & cos\alpha \end{matrix})(\begin{matrix}x_{i} \\ y_{i} \end{matrix}) = (\begin{matrix}cos\alpha x_i - sin\alpha y_i \\ sin\alpha x_i + cos\alpha y_i \end{matrix}) (xi+1yi+1)=(cosαsinα−sinαcosα)(xiyi)=(cosαxi−sinαyisinαxi+cosαyi)

'''圆内接正多边形逼近法'''
def drawArc_InscribedRegularPolygonApproximate(grid, R):Alpha = 1/RcosAlpha, sinAlpha = cos(Alpha), sin(Alpha)x, y = R, 0while x >= y:drawPixel_symmetry8(int(x+0.5), int(y+0.5), 1, grid)x = cosAlpha * x - sinAlpha * yy = sinAlpha * x + cosAlpha * y