本例子是简单的在WinForm程序中实现在坐标系中绘制直线方程，抛物线方程，点。重新学习解析几何方面的知识。仅供学习分享使用，如有不足之处，还请指正。
涉及知识点：
- 直线方程的表达方式：一般表达式Ax By C=0
- 抛物线表达式：y=Ax2 Bx C
- 坐标转换：由于WinForm中的坐标原点是左上角，数学二维坐标系的原点是在中间，所以需要转换
- 单位转换：WinForm的单位是Pixls，但坐标系的单位不是，需要进行缩放。
- 画图方法：程序中使用GDI 进行画图。
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效果图
如下：
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多边形扩展
【如下图所示】
//求多边形对应的边的平行线，然后再求相邻平行线的交点，连起来即是扩展多边形
核心算法
主要代码如下：
【方程类】
代码语言：javascript复制 1 using System;
2 using System.Collections.Generic;
3 using System.Linq;
4 using System.Text;
5
6 namespace DemoGeometry
7 {
8 /// <summary>
9 /// 方程基类
10 /// </summary>
11 public abstract class Equation
12 {
13 public int A { get; set; }
14 public int B { get; set; }
15 public int C { get; set; }
16
17 /// <summary>
18 /// 判断是否有效
19 /// </summary>
20 /// <returns></returns>
21 public abstract bool IsValid();
22
23 /// <summary>
24 /// 通过Y值获取x值
25 /// </summary>
26 /// <param name="y"></param>
27 /// <returns></returns>
28 public abstract float GetValueFromY(float y);
29 /// <summary>
30 /// 通过X获取Y值
31 /// </summary>
32 /// <param name="x"></param>
33 /// <returns></returns>
34 public abstract float GetValueFromX(float x);
35 }
36
37 /// <summary>
38 /// 直线方程类一般式:Ax By C=0(A、B不同时为0)【适用于所有直线】
39 /// </summary>
40 public class LinearEquation:Equation
41 {
42 /// <summary>
43 /// 通过X值得到Y值
44 /// </summary>
45 /// <param name="x"></param>
46 /// <returns></returns>
47 public override float GetValueFromX(float x)
48 {
49 if (B == 0)
50 {
51 return float.MaxValue;
52 }
53 return -A * x * 1.0f / B - C * 1.0f / B;
54 }
55
56 public override float GetValueFromY(float y)
57 {
58 if (A == 0)
59 {
60 return float.MaxValue;
61 }
62 return -B * y * 1.0f / A - C * 1.0f / A;
63 }
64
65 /// <summary>
66 /// 判断是否有效方程
67 /// </summary>
68 /// <returns></returns>
69 public override bool IsValid()
70 {
71 bool flag = true;
72 if (A == 0 && B == 0)
73 {
74 flag = false;
75 }
76 return flag;
77 }
78
79 public override string ToString()
80 {
81 return string.Format("{0}x {1}y {2}=0", A, B, C);
82 }
83 }
84
85 /// <summary>
86 /// 抛物线方程表达式 y=ax2 bx c
87 /// </summary>
88 public class ParabolicEquation:Equation
89 {
90
91 /// <summary>
92 /// 判断是否有效的方程
93 /// </summary>
94 /// <returns></returns>
95 public override bool IsValid()
96 {
97 //A 不得等于0
98 return A != 0;
99 }
100
101 /// <summary>
102 /// 通过X值得到Y值
103 /// </summary>
104 /// <param name="x"></param>
105 /// <returns></returns>
106 public override float GetValueFromX(float x)
107 {
108 double y = A * Math.Pow(x, 2) B * x C;
109 return float.Parse(y.ToString());
110 }
111
112 public override float GetValueFromY(float y)
113 {
114 return 0.0f;
115 }
116 }
117 }【控件类】
代码语言：javascript复制 1 using System;
2 using System.Collections.Generic;
3 using System.ComponentModel;
4 using System.Drawing;
5 using System.Data;
6 using System.Linq;
7 using System.Text;
8 using System.Windows.Forms;
9
10 namespace DemoGeometry
11 {
12 /// <summary>
13 /// 坐标系控件
14 /// </summary>
15 public partial class AxisControl : UserControl
16 {
17 #region 属性
18
19 private Axis _AxisX = new Axis() { XName = "x" };
20 /// <summary>
21 /// X轴
22 /// </summary>
23 public Axis AxisX { get { return this._AxisX; } }
24
25 private Axis _AxisY = new Axis() { XName = "y" };
26 /// <summary>
27 /// Y轴
28 /// </summary>
29 public Axis AxisY { get { return this._AxisY; } }
30
31 /// <summary>
32 /// 边界留空白
33 /// </summary>
34 private int bound = 10;
35
36 public int Bound
37 {
38 get { return bound; }
39 set { bound = value; }
40 }
41
42 /// <summary>
43 /// 表示单位，10个像素表示1
44 /// </summary>
45 private int unit = 30;
46
47 public int Unit
48 {
49 get { return unit; }
50 set { unit = value; }
51 }
52
53 /// <summary>
54 /// 文本字体
55 /// </summary>
56 private Font t_Font = new Font("Arial", 10F);
57
58 private PointF center;
59
60 private int index = 0;
61
62 private int lineWidth = 2;
63
64 #endregion
65
66 public AxisControl()
67 {
68 InitializeComponent();
69 }
70
71 private void InitInfo() {
72 //绘制坐标轴
73 var width = this.Width * 1.0f;
74 var height = this.Height * 1.0f;
75 center = new PointF(width / 2, height / 2);
76 }
77
78 /// <summary>
79 /// 重绘界面函数
80 /// </summary>
81 /// <param name="e"></param>
82 protected override void OnPaint(PaintEventArgs e)
83 {
84 base.OnPaint(e);
85 InitInfo();
86 Graphics g = e.Graphics;
87 var width = this.Width * 1.0f;
88 var height = this.Height * 1.0f;
89 Pen pen = new Pen(Color.Black);
90 var left = new PointF(bound,center.Y);//
91 var right = new PointF(width - bound, center.Y);
92 var bottom = new PointF(center.X, height - bound);
93 var top = new PointF(center.X, bound);
94 g.DrawString("0", t_Font, Brushes.Black, new PointF(center.X bound/2, center.Y bound/2));
95 //画X轴,X轴的箭头
96 g.DrawLine(pen, left, right);
97 g.DrawLine(pen, new PointF(right.X - bound/2, right.Y - bound/2), right);
98 g.DrawLine(pen, new PointF(right.X - bound/2, right.Y bound/2), right);
99 var xName = string.IsNullOrEmpty(this._AxisX.XName) ? "x" : this._AxisX.XName;
100 g.DrawString(xName, t_Font, Brushes.Black, new PointF(right.X - bound, right.Y 2 * bound));
101 //绘制X轴的刻度
102 var xMax = Math.Floor((right.X - left.X) / (2*unit));//
103 this._AxisX.Max = int.Parse( xMax.ToString());
104 for (var i = 0; i < xMax; i ) {
105 //正刻度
106 g.DrawLine(pen, new PointF(center.X (i 1) * unit, center.Y), new PointF(center.X (i 1) * unit, center.Y - 2));
107 g.DrawString((i 1).ToString(), t_Font, Brushes.Black, new PointF(center.X (i 1) * unit, center.Y 2));
108 //负刻度
109 g.DrawLine(pen, new PointF(center.X - (i 1) * unit, center.Y), new PointF(center.X - (i 1) * unit, center.Y - 2));
110 g.DrawString("-" (i 1).ToString(), t_Font, Brushes.Black, new PointF(center.X - (i 1) * unit, center.Y 2));
111 }
112 //画Y轴，Y轴的箭头
113 g.DrawLine(pen, bottom, top);
114 g.DrawLine(pen, new PointF(top.X - bound/2, top.Y bound/2), top);
115 g.DrawLine(pen, new PointF(top.X bound/2, top.Y bound/2), top);
116 var yName = string.IsNullOrEmpty(_AxisY.XName) ? "y" : _AxisY.XName;
117 g.DrawString(AxisY.XName, t_Font, Brushes.Black, new PointF(top.X 2 * bound, top.Y - bound));
118 //绘制Y轴的刻度
119 var yMax = Math.Floor((bottom.Y - top.Y) / (2 * unit));//
120 this._AxisY.Max = int.Parse(yMax.ToString());
121 for (var i = 0; i < yMax; i )
122 {
123 //正刻度
124 g.DrawLine(pen, new PointF(center.X , center.Y- (i 1) * unit), new PointF(center.X 2 , center.Y- (i 1) * unit ));
125 g.DrawString((i 1).ToString(), t_Font, Brushes.Black, new PointF(center.X 2 , center.Y- (i 1) * unit));
126 //负刻度
127 g.DrawLine(pen, new PointF(center.X, center.Y (i 1) * unit), new PointF(center.X 2, center.Y (i 1) * unit ));
128 g.DrawString("-" (i 1).ToString(), t_Font, Brushes.Black, new PointF(center.X 2 , center.Y (i 1) * unit ));
129 }
130 }
131
132 /// <summary>
133 /// 判断直线方程是否在坐标轴范围内
134 /// </summary>
135 /// <param name="linear"></param>
136 /// <returns></returns>
137 public bool CheckLineIsValid(LinearEquation linear)
138 {
139 bool flagX = false;
140 bool flagY = false;
141 var y = linear.GetValueFromX(0f);
142
143 //判断y坐标的值有没有越界
144 if (y == float.MaxValue)
145 {
146 //表示是垂直于x轴的直线
147 var x0 = -linear.C*1.0f / linear.A;
148 if (x0 >= 0 - this._AxisX.Max && x0 < this._AxisX.Max)
149 {
150 flagY = true;
151 }
152 else
153 {
154 flagY = false;
155 }
156
157 }
158 else
159 {
160 if (y <= this._AxisY.Max && y >= 0 - this._AxisY.Max)
161 {
162 flagY = true;
163 }
164 else
165 {
166 flagY = false;
167 }
168 }
169 //判断x坐标的值
170 var x = linear.GetValueFromY(0f);
171 if (x == float.MaxValue)
172 {
173 var y0 = -linear.C*1.0f / linear.B;
174
175 if (y0 <= this._AxisY.Max && y0 >= 0 - this._AxisY.Max)
176 {
177 flagX = true;
178 }
179 else
180 {
181 flagX = false;
182 }
183 }
184 else
185 {
186 if (x <= this._AxisX.Max && x >= 0 - this._AxisX.Max)
187 {
188 flagX = true;
189 }
190 else
191 {
192 flagX = false;
193 }
194 }
195
196 return flagX && flagY;//只有x,y都满足条件，才是有效的
197 }
198
199 /// <summary>
200 /// 判断点是否在坐标轴范围内
201 /// </summary>
202 /// <returns></returns>
203 public bool CheckPointIsValid(PointF point)
204 {
205 bool flagX = false;
206 bool flagY = false;
207 if (point.X <= this._AxisX.Max && point.X >= 0 - this._AxisX.Max)
208 {
209 flagX = true;
210 }
211 if (point.Y <= this._AxisY.Max && point.Y >= 0 - this._AxisY.Max)
212 {
213 flagY = true;
214 }
215 return flagX && flagY;
216 }
217
218 /// <summary>
219 /// 检查抛物线方程是否有效
220 /// </summary>
221 /// <param name="parabolic"></param>
222 /// <returns></returns>
223 public bool CheckParabolicIsValid(ParabolicEquation parabolic) {
224 List<PointF> lstPoint = GetPointFromEquation(parabolic);
225 if (lstPoint.Count > 2)
226 {
227 return true;
228 }
229 else {
230 return false;
231 }
232 }
233
234 /// <summary>
235 /// 将刻度转换成像素
236 /// </summary>
237 public List<PointF> ConvertScaleToPixls(List<PointF> lstScale)
238 {
239 List<PointF> lstPixls = new List<PointF>();
240 if (lstScale != null && lstScale.Count > 0) {
241 var p = lstScale.Select(s => new PointF(center.X s.X * unit, center.Y - s.Y * unit));
242 lstPixls = p.ToList();
243 }
244 return lstPixls;
245 }
246
247 /// <summary>
248 /// 转换刻度到像素
249 /// </summary>
250 /// <param name="s"></param>
251 /// <returns></returns>
252 public PointF ConvertScaleToPixls(PointF s)
253 {
254 return new PointF(center.X s.X * unit, center.Y - s.Y * unit);
255 }
256
257 /// <summary>
258 /// 生成直线
259 /// </summary>
260 /// <param name="linear"></param>
261 public bool GenerateLinear(LinearEquation linear) {
262
263 Color lineColor = Color.Blue;//线条的颜色
264 Graphics g = this.CreateGraphics();
265 //分别获取两个端点的值，连成线即可
266 var x1 = this._AxisX.Max;
267 var y2 = this._AxisY.Max;
268 var x3 = 0 - this._AxisX.Max;
269 var y4 = 0 - this._AxisY.Max;
270 var y1 = linear.GetValueFromX(x1);
271 var x2 = linear.GetValueFromY(y2);
272 var y3 = linear.GetValueFromX(x3);
273 var x4 = linear.GetValueFromY(y4);
274 var point1 = new PointF(x1, y1);
275 var point2 = new PointF(x2, y2);
276 var point3 = new PointF(x3, y3);
277 var point4 = new PointF(x4, y4);
278 List<PointF> lstTmp = new List<PointF>() { point1, point2, point3, point4 };
279 List<PointF> lstPoint=new List<PointF>();
280 foreach (PointF point in lstTmp)
281 {
282 //判断点是否合理
283 if (CheckPointIsValid(point))
284 {
285 lstPoint.Add(point);
286 }
287 }
288 if (lstPoint.Count() < 2) {
289 //如果点的个数小于2，不能绘制直线
290 return false;
291 }
292 //将刻度点，转换成像素点
293 List<PointF> lstPixlsPoint = ConvertScaleToPixls(lstPoint);
294 g.DrawLine(new Pen(lineColor,lineWidth),lstPixlsPoint[0],lstPixlsPoint[1]);
295 g.DrawString(string.Format("L{0}", index), t_Font, Brushes.Black, new PointF(lstPixlsPoint[1].X 2, lstPixlsPoint[1].Y - 2));
296 this.lblInfo.Text = string.Format("L{0}:{1}x {2}y {3}=0 ; ", index, linear.A, linear.B, linear.C);
297 index ;
298 return true;
299 }
300
301 /// <summary>
302 /// 生成点
303 /// </summary>
304 /// <param name="point"></param>
305 /// <returns></returns>
306 public bool GeneratePoint(PointF point)
307 {
308 Graphics g = this.CreateGraphics();
309 PointF p = ConvertScaleToPixls(point);
310 g.FillEllipse(Brushes.Red, p.X, p.Y, 4, 4);
311 g.DrawString(string.Format("P{0}", index), t_Font, Brushes.Black, new PointF(p.X 4, p.Y - 4));
312 this.lblInfo.Text = string.Format("P{0}:({1},{2}) ; ", index, point.X, point.Y);
313 index ;
314 return true;
315 }
316
317 public bool GenerateParabolic(ParabolicEquation parabolic)
318 {
319 List<PointF> lstPoint = GetPointFromEquation(parabolic);
320 //将刻度点，转换成像素点
321 List<PointF> lstPixlsPoint = ConvertScaleToPixls(lstPoint);
322 Color lineColor = Color.SeaGreen;//线条的颜色
323 Graphics g = this.CreateGraphics();
324 g.DrawCurve(new Pen(lineColor,lineWidth), lstPixlsPoint.ToArray());
325 g.DrawString(string.Format("P{0}", index), t_Font, Brushes.Black, new PointF(lstPixlsPoint[1].X 2, lstPixlsPoint[1].Y - 2));
326 this.lblInfo.Text = string.Format("P{0}:y={1}x2 {2}x {3} ; ", index, parabolic.A, parabolic.B, parabolic.C);
327 index ;
328 return true;
329 }
330
331 /// <summary>
332 /// 从抛物线方程中取点值
333 /// </summary>
334 /// <param name="parabolic"></param>
335 /// <returns></returns>
336 public List<PointF> GetPointFromEquation(ParabolicEquation parabolic) {
337 List<PointF> lstPoint = new List<PointF>();
338 //从坐标轴最小值开始，隔0.5 取一个
339 int j=0;
340 for (float i = 0 - this._AxisX.Max; i <= this._AxisX.Max; i = i 0.5f)
341 {
342 PointF p = new PointF(i, parabolic.GetValueFromX(i));
343 //再判断点是否在坐标轴内
344 if (CheckPointIsValid(p) && (j==0 || lstPoint[j-1].X==i-0.5f))
345 {
346 lstPoint.Add(p);//抛物线内的点应该是连续的
347 j ;
348 }
349 }
350 return lstPoint;
351 }
352
353 /// <summary>
354 /// 清除已经画上去的线条
355 /// </summary>
356 /// <returns></returns>
357 public bool Clear() {
358 Graphics g = this.CreateGraphics();
359 g.Clear(Color.White);
360 this.lblInfo.Text = "";
361 this.Refresh();//重新刷新界面，清除已经画上去的线条
362 index = 0;
363 return true;
364 }
365 }
366
367 /// <summary>
368 /// 坐标轴描述
369 /// </summary>
370 public class Axis{
371
372 /// <summary>
373 /// 刻度表示最大值
374 /// </summary>
375 public int Max { get; set; }
376
377 /// <summary>
378 /// 名称
379 /// </summary>
380 public string XName{get;set;}
381
382 ///// <summary>
383 ///// 间隔
384 ///// </summary>
385 //public int Interval{get;set;}
386 }
387 }【主界面类】
代码语言：javascript复制 1 using System;
2 using System.Collections.Generic;
3 using System.ComponentModel;
4 using System.Data;
5 using System.Drawing;
6 using System.Linq;
7 using System.Text;
8 using System.Windows.Forms;
9
10 namespace DemoGeometry
11 {
12 public partial class MainForm : Form
13 {
14 public MainForm()
15 {
16 InitializeComponent();
17 }
18
19 private void btnLine_Click(object sender, EventArgs e)
20 {
21 if (this.linearControl1.IsValid())
22 {
23 var a = this.linearControl1.A;
24 var b = this.linearControl1.B;
25 var c = this.linearControl1.C;
26 //判断方程的参数，是否有效
27 LinearEquation linear = new LinearEquation() { A = a, B = b, C = c };
28 if (!linear.IsValid())
29 {
30 MessageBox.Show("输入的方程参数无效");
31 return;
32 }
33 if (!this.axisControl1.CheckLineIsValid(linear))
34 {
35 MessageBox.Show("输入的方程不在坐标轴内");
36 return;
37 }
38 bool flag = this.axisControl1.GenerateLinear(linear);
39 if (!flag)
40 {
41 MessageBox.Show("生成直线失败");
42 return;
43 }
44 }
45 }
46
47 private void btnClear_Click(object sender, EventArgs e)
48 {
49 this.axisControl1.Clear();
50 }
51
52 private void btnPoint_Click(object sender, EventArgs e)
53 {
54 if (this.pointControl1.IsValid())
55 {
56 float x = this.pointControl1.X;
57 float y = this.pointControl1.Y;
58 PointF point = new PointF(x, y);
59 if (!this.axisControl1.CheckPointIsValid(point)) {
60 MessageBox.Show("输入的点不在坐标轴内");
61 return;
62 }
63 bool flag = this.axisControl1.GeneratePoint(point);
64 if (!flag)
65 {
66 MessageBox.Show("生成点失败");
67 return;
68 }
69 }
70
71 }
72
73 private void btnParabolic_Click(object sender, EventArgs e)
74 {
75 if (this.parabolicControl1.IsValid())
76 {
77 var a = this.parabolicControl1.A;
78 var b = this.parabolicControl1.B;
79 var c = this.parabolicControl1.C;
80 //判断方程的参数，是否有效
81 ParabolicEquation parabolic = new ParabolicEquation() { A = a, B = b, C = c };
82 if (!parabolic.IsValid())
83 {
84 MessageBox.Show("输入的方程参数无效");
85 return;
86 }
87 if (!this.axisControl1.CheckParabolicIsValid(parabolic))
88 {
89 MessageBox.Show("输入的方程不在坐标轴内");
90 return;
91 }
92 bool flag = this.axisControl1.GenerateParabolic(parabolic);
93 if (!flag)
94 {
95 MessageBox.Show("生成抛物线失败");
96 return;
97 }
98 }
99 }
100 }
101 }【扩展多边形代码】
代码语言：javascript复制1 /// <summary>
2 /// 生成多边形
3 /// </summary>
4 /// <param name="lstPoints"></param>
5 /// <returns></returns>
6 public bool GeneratePolygon(List<PointF> lstPoints ,bool isOriginal=true) {
7
8 Color lineColor = Color.Red;//线条的颜色
9 if (isOriginal)
10 {
11 lineColor = Color.Red;
12 }
13 else {
14 lineColor = Color.Blue;
15 }
16
17 Graphics g = this.CreateGraphics();
18 //画点
19 foreach (var p in lstPoints) {
20 this.GeneratePoint(p);
21 }
22 //将刻度点，转换成像素点
23 List<PointF> lstPixlsPoint = ConvertScaleToPixls(lstPoints);
24 //绘制多边形
25 g.DrawPolygon(new Pen(lineColor,2), lstPixlsPoint.ToArray());
26 return true;
27 }
28
29 /// <summary>
30 /// 扩展多边形，即在现有多边形的基础上进行扩展
31 /// </summary>
32 /// <param name="lstPoints"></param>
33 /// <returns></returns>
34 public bool GenerateExpandPolygon(List<PointF> lstPoints,PointF center,float distance) {
35 //1.求多边形对应的外扩平行斜线
36 List<LinearEquation> lstLines = new List<LinearEquation>();
37 int len = lstPoints.Count();
38 for (int i = 0; i < len; i ) {
39 var p0 = lstPoints[i];//第i个元素
40 var p1 = lstPoints[(i 1) % len];//第i 1个元素
41 LinearEquation linearEquation = new LinearEquation();
42 if (p0.X == p1.X)
43 {
44 //垂直于x轴，没有斜率
45 linearEquation.A = 1;
46 linearEquation.B = 0;
47 if (p0.X > center.X)
48 {
49 linearEquation.C = -(p0.X distance);
50 }
51 else {
52 linearEquation.C = -(p0.X - distance);
53 }
54
55 }
56 else if (p0.Y == p1.Y)
57 {
58 //垂直于y轴，斜率为0
59 linearEquation.A = 0;
60 linearEquation.B = 1;
61 if (p0.Y > center.Y)
62 {
63 linearEquation.C = -(p0.Y distance);
64 }
65 else {
66 linearEquation.C = -(p0.Y - distance);
67 }
68 }
69 else {
70 //先求两点对应的点斜式方程y=kx b
71 float k = (p0.Y - p1.Y) / (p0.X - p1.X);
72 float b = p0.Y - k * p0.X;
73 //求出平行线对应的b即可。
74 float b_center = center.Y - k * center.X;
75 linearEquation.A = k;
76 linearEquation.B = -1;//如果此处为-1,则C=b
77
78 if (b > b_center)
79 {
80 //如果在原点上方
81
82 linearEquation.C =(b (float) Math.Abs((distance/(Math.Sin(-Math.PI/2-Math.Atan(k))))));
83 }
84 else {
85 //如果在原点下方
86 linearEquation.C = (b - (float)Math.Abs((distance / (Math.Sin(-Math.PI / 2 - Math.Atan(k))))));
87 }
88 }
89 //this.GenerateLinear(linearEquation);
90 lstLines.Add(linearEquation);
91 }
92 List<PointF> lstNewPoints = new List<PointF>();
93 //2.求相邻外扩平行斜线的交点
94 for (int i = 0; i < len; i ) {
95 var line0 = lstLines[i];
96 var line1 = lstLines[(i 1) % len];
97 float x = 0.0f;
98 float y = 0.0f;
99 if (line0.A == 0.0f)
100 {
101 y = -line0.C / line0.B;
102 x= line1.GetValueFromY(y);
103 }
104 else if (line0.B == 0.0f)
105 {
106 x = -line0.C / line0.A;
107 y = line1.GetValueFromX(x);
108 }
109 else if (line1.A == 0.0f)
110 {
111 y = -line1.C / line1.B;
112 x = line0.GetValueFromY(y);
113 }
114 else if (line1.B == 0.0f)
115 {
116 x = -line1.C / line1.A;
117 y = line0.GetValueFromX(x);
118 }
119 else {
120 //两个都有斜率的直线的交点
121 float k0 = -line0.A / line0.B;
122 float k1 = -line1.A / line1.B;
123 float b0 = -line0.C / line0.B;
124 float b1 = -line1.C / line1.B;
125 x = (b1 - b0) / (k0 - k1);
126 y = line0.GetValueFromX(x);
127 }
128 lstNewPoints.Add(new PointF(x, y));
129 }
130 this.GeneratePolygon(lstNewPoints, false);
131 //
132 return true;
133 }备注：源码 点击下方链接
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转载于:https://www.cnblogs.com/hsiang/p/6294864.html
相关资源：DirectX修复工具V4.0增强版
