C# 简单实现直线方程，抛物线方程
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：
---------------------------------------------------------------------------------------------------------
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】
using System;
using System.Collections.Generic;
using System.Linq;
using System.Text; namespace DemoGeometry
{
/// <summary>
/// 方程基类
/// </summary>
public abstract class Equation
{
public int A { get; set; }
public int B { get; set; }
public int C { get; set; } /// <summary>
/// 判断是否有效
/// </summary>
/// <returns></returns>
public abstract bool IsValid(); /// <summary>
/// 通过Y值获取x值
/// </summary>
/// <param name="y"></param>
/// <returns></returns>
public abstract float GetValueFromY(float y);
/// <summary>
/// 通过X获取Y值
/// </summary>
/// <param name="x"></param>
/// <returns></returns>
public abstract float GetValueFromX(float x);
} /// <summary>
/// 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】
/// </summary>
public class LinearEquation:Equation
{
/// <summary>
/// 通过X值得到Y值
/// </summary>
/// <param name="x"></param>
/// <returns></returns>
public override float GetValueFromX(float x)
{
if (B == )
{
return float.MaxValue;
}
return -A * x * 1.0f / B - C * 1.0f / B;
} public override float GetValueFromY(float y)
{
if (A == )
{
return float.MaxValue;
}
return -B * y * 1.0f / A - C * 1.0f / A;
} /// <summary>
/// 判断是否有效方程
/// </summary>
/// <returns></returns>
public override bool IsValid()
{
bool flag = true;
if (A == && B == )
{
flag = false;
}
return flag;
} public override string ToString()
{
return string.Format("{0}x+{1}y+{2}=0", A, B, C);
}
} /// <summary>
/// 抛物线方程表达式 y=ax2+bx+c
/// </summary>
public class ParabolicEquation:Equation
{ /// <summary>
/// 判断是否有效的方程
/// </summary>
/// <returns></returns>
public override bool IsValid()
{
//A 不得等于0
return A != ;
} /// <summary>
/// 通过X值得到Y值
/// </summary>
/// <param name="x"></param>
/// <returns></returns>
public override float GetValueFromX(float x)
{
double y = A * Math.Pow(x, ) + B * x + C;
return float.Parse(y.ToString());
} public override float GetValueFromY(float y)
{
return 0.0f;
}
}
}
【控件类】
using System;
using System.Collections.Generic;
using System.ComponentModel;
using System.Drawing;
using System.Data;
using System.Linq;
using System.Text;
using System.Windows.Forms; namespace DemoGeometry
{
/// <summary>
/// 坐标系控件
/// </summary>
public partial class AxisControl : UserControl
{
#region 属性 private Axis _AxisX = new Axis() { XName = "x" };
/// <summary>
/// X轴
/// </summary>
public Axis AxisX { get { return this._AxisX; } } private Axis _AxisY = new Axis() { XName = "y" };
/// <summary>
/// Y轴
/// </summary>
public Axis AxisY { get { return this._AxisY; } } /// <summary>
/// 边界留空白
/// </summary>
private int bound = ; public int Bound
{
get { return bound; }
set { bound = value; }
} /// <summary>
/// 表示单位，10个像素表示1
/// </summary>
private int unit = ; public int Unit
{
get { return unit; }
set { unit = value; }
} /// <summary>
/// 文本字体
/// </summary>
private Font t_Font = new Font("Arial", 10F); private PointF center; private int index = ; private int lineWidth = ; #endregion public AxisControl()
{
InitializeComponent();
} private void InitInfo() {
//绘制坐标轴
var width = this.Width * 1.0f;
var height = this.Height * 1.0f;
center = new PointF(width / , height / );
} /// <summary>
/// 重绘界面函数
/// </summary>
/// <param name="e"></param>
protected override void OnPaint(PaintEventArgs e)
{
base.OnPaint(e);
InitInfo();
Graphics g = e.Graphics;
var width = this.Width * 1.0f;
var height = this.Height * 1.0f;
Pen pen = new Pen(Color.Black);
var left = new PointF(bound,center.Y);//
var right = new PointF(width - bound, center.Y);
var bottom = new PointF(center.X, height - bound);
var top = new PointF(center.X, bound);
g.DrawString("", t_Font, Brushes.Black, new PointF(center.X + bound/, center.Y+bound/));
//画X轴,X轴的箭头
g.DrawLine(pen, left, right);
g.DrawLine(pen, new PointF(right.X - bound/, right.Y - bound/), right);
g.DrawLine(pen, new PointF(right.X - bound/, right.Y + bound/), right);
var xName = string.IsNullOrEmpty(this._AxisX.XName) ? "x" : this._AxisX.XName;
g.DrawString(xName, t_Font, Brushes.Black, new PointF(right.X - bound, right.Y + * bound));
//绘制X轴的刻度
var xMax = Math.Floor((right.X - left.X) / (*unit));//
this._AxisX.Max = int.Parse( xMax.ToString());
for (var i = ; i < xMax; i++) {
//正刻度
g.DrawLine(pen, new PointF(center.X + (i + ) * unit, center.Y), new PointF(center.X + (i + ) * unit, center.Y - ));
g.DrawString((i + ).ToString(), t_Font, Brushes.Black, new PointF(center.X + (i + ) * unit, center.Y+));
//负刻度
g.DrawLine(pen, new PointF(center.X - (i + ) * unit, center.Y), new PointF(center.X - (i + ) * unit, center.Y - ));
g.DrawString("-"+(i + ).ToString(), t_Font, Brushes.Black, new PointF(center.X - (i + ) * unit, center.Y + ));
}
//画Y轴，Y轴的箭头
g.DrawLine(pen, bottom, top);
g.DrawLine(pen, new PointF(top.X - bound/, top.Y + bound/), top);
g.DrawLine(pen, new PointF(top.X + bound/, top.Y + bound/), top);
var yName = string.IsNullOrEmpty(_AxisY.XName) ? "y" : _AxisY.XName;
g.DrawString(AxisY.XName, t_Font, Brushes.Black, new PointF(top.X + * bound, top.Y - bound));
//绘制Y轴的刻度
var yMax = Math.Floor((bottom.Y - top.Y) / ( * unit));//
this._AxisY.Max = int.Parse(yMax.ToString());
for (var i = ; i < yMax; i++)
{
//正刻度
g.DrawLine(pen, new PointF(center.X , center.Y- (i + ) * unit), new PointF(center.X+ , center.Y- (i + ) * unit ));
g.DrawString((i + ).ToString(), t_Font, Brushes.Black, new PointF(center.X+ , center.Y- (i + ) * unit));
//负刻度
g.DrawLine(pen, new PointF(center.X, center.Y + (i + ) * unit), new PointF(center.X+ , center.Y + (i + ) * unit ));
g.DrawString("-" + (i + ).ToString(), t_Font, Brushes.Black, new PointF(center.X+ , center.Y+ (i + ) * unit ));
}
} /// <summary>
/// 判断直线方程是否在坐标轴范围内
/// </summary>
/// <param name="linear"></param>
/// <returns></returns>
public bool CheckLineIsValid(LinearEquation linear)
{
bool flagX = false;
bool flagY = false;
var y = linear.GetValueFromX(0f); //判断y坐标的值有没有越界
if (y == float.MaxValue)
{
//表示是垂直于x轴的直线
var x0 = -linear.C*1.0f / linear.A;
if (x0 >= - this._AxisX.Max && x0 < this._AxisX.Max)
{
flagY = true;
}
else
{
flagY = false;
} }
else
{
if (y <= this._AxisY.Max && y >= - this._AxisY.Max)
{
flagY = true;
}
else
{
flagY = false;
}
}
//判断x坐标的值
var x = linear.GetValueFromY(0f);
if (x == float.MaxValue)
{
var y0 = -linear.C*1.0f / linear.B; if (y0 <= this._AxisY.Max && y0 >= - this._AxisY.Max)
{
flagX = true;
}
else
{
flagX = false;
}
}
else
{
if (x <= this._AxisX.Max && x >= - this._AxisX.Max)
{
flagX = true;
}
else
{
flagX = false;
}
} return flagX && flagY;//只有x,y都满足条件，才是有效的
} /// <summary>
/// 判断点是否在坐标轴范围内
/// </summary>
/// <returns></returns>
public bool CheckPointIsValid(PointF point)
{
bool flagX = false;
bool flagY = false;
if (point.X <= this._AxisX.Max && point.X >= - this._AxisX.Max)
{
flagX = true;
}
if (point.Y <= this._AxisY.Max && point.Y >= - this._AxisY.Max)
{
flagY = true;
}
return flagX && flagY;
} /// <summary>
/// 检查抛物线方程是否有效
/// </summary>
/// <param name="parabolic"></param>
/// <returns></returns>
public bool CheckParabolicIsValid(ParabolicEquation parabolic) {
List<PointF> lstPoint = GetPointFromEquation(parabolic);
if (lstPoint.Count > )
{
return true;
}
else {
return false;
}
} /// <summary>
/// 将刻度转换成像素
/// </summary>
public List<PointF> ConvertScaleToPixls(List<PointF> lstScale)
{
List<PointF> lstPixls = new List<PointF>();
if (lstScale != null && lstScale.Count > ) {
var p = lstScale.Select(s => new PointF(center.X + s.X * unit, center.Y - s.Y * unit));
lstPixls = p.ToList();
}
return lstPixls;
} /// <summary>
/// 转换刻度到像素
/// </summary>
/// <param name="s"></param>
/// <returns></returns>
public PointF ConvertScaleToPixls(PointF s)
{
return new PointF(center.X + s.X * unit, center.Y - s.Y * unit);
} /// <summary>
/// 生成直线
/// </summary>
/// <param name="linear"></param>
public bool GenerateLinear(LinearEquation linear) { Color lineColor = Color.Blue;//线条的颜色
Graphics g = this.CreateGraphics();
//分别获取两个端点的值，连成线即可
var x1 = this._AxisX.Max;
var y2 = this._AxisY.Max;
var x3 = - this._AxisX.Max;
var y4 = - this._AxisY.Max;
var y1 = linear.GetValueFromX(x1);
var x2 = linear.GetValueFromY(y2);
var y3 = linear.GetValueFromX(x3);
var x4 = linear.GetValueFromY(y4);
var point1 = new PointF(x1, y1);
var point2 = new PointF(x2, y2);
var point3 = new PointF(x3, y3);
var point4 = new PointF(x4, y4);
List<PointF> lstTmp = new List<PointF>() { point1, point2, point3, point4 };
List<PointF> lstPoint=new List<PointF>();
foreach (PointF point in lstTmp)
{
//判断点是否合理
if (CheckPointIsValid(point))
{
lstPoint.Add(point);
}
}
if (lstPoint.Count() < ) {
//如果点的个数小于2，不能绘制直线
return false;
}
//将刻度点，转换成像素点
List<PointF> lstPixlsPoint = ConvertScaleToPixls(lstPoint);
g.DrawLine(new Pen(lineColor,lineWidth),lstPixlsPoint[],lstPixlsPoint[]);
g.DrawString(string.Format("L{0}", index), t_Font, Brushes.Black, new PointF(lstPixlsPoint[].X + , lstPixlsPoint[].Y - ));
this.lblInfo.Text += string.Format("L{0}:{1}x+{2}y+{3}=0 ; ", index, linear.A, linear.B, linear.C);
index++;
return true;
} /// <summary>
/// 生成点
/// </summary>
/// <param name="point"></param>
/// <returns></returns>
public bool GeneratePoint(PointF point)
{
Graphics g = this.CreateGraphics();
PointF p = ConvertScaleToPixls(point);
g.FillEllipse(Brushes.Red, p.X, p.Y, , );
g.DrawString(string.Format("P{0}", index), t_Font, Brushes.Black, new PointF(p.X + , p.Y - ));
this.lblInfo.Text += string.Format("P{0}:({1},{2}) ; ", index, point.X, point.Y);
index++;
return true;
} public bool GenerateParabolic(ParabolicEquation parabolic)
{
List<PointF> lstPoint = GetPointFromEquation(parabolic);
//将刻度点，转换成像素点
List<PointF> lstPixlsPoint = ConvertScaleToPixls(lstPoint);
Color lineColor = Color.SeaGreen;//线条的颜色
Graphics g = this.CreateGraphics();
g.DrawCurve(new Pen(lineColor,lineWidth), lstPixlsPoint.ToArray());
g.DrawString(string.Format("P{0}", index), t_Font, Brushes.Black, new PointF(lstPixlsPoint[].X + , lstPixlsPoint[].Y - ));
this.lblInfo.Text += string.Format("P{0}:y={1}x2+{2}x+{3} ; ", index, parabolic.A, parabolic.B, parabolic.C);
index++;
return true;
} /// <summary>
/// 从抛物线方程中取点值
/// </summary>
/// <param name="parabolic"></param>
/// <returns></returns>
public List<PointF> GetPointFromEquation(ParabolicEquation parabolic) {
List<PointF> lstPoint = new List<PointF>();
//从坐标轴最小值开始，隔0.5 取一个
int j=;
for (float i = - this._AxisX.Max; i <= this._AxisX.Max; i = i + 0.5f)
{
PointF p = new PointF(i, parabolic.GetValueFromX(i));
//再判断点是否在坐标轴内
if (CheckPointIsValid(p) && (j== || lstPoint[j-].X==i-0.5f))
{
lstPoint.Add(p);//抛物线内的点应该是连续的
j++;
}
}
return lstPoint;
} /// <summary>
/// 清除已经画上去的线条
/// </summary>
/// <returns></returns>
public bool Clear() {
Graphics g = this.CreateGraphics();
g.Clear(Color.White);
this.lblInfo.Text = "";
this.Refresh();//重新刷新界面，清除已经画上去的线条
index = ;
return true;
}
} /// <summary>
/// 坐标轴描述
/// </summary>
public class Axis{ /// <summary>
/// 刻度表示最大值
/// </summary>
public int Max { get; set; } /// <summary>
/// 名称
/// </summary>
public string XName{get;set;} ///// <summary>
///// 间隔
///// </summary>
//public int Interval{get;set;}
}
}
【主界面类】
using System;
using System.Collections.Generic;
using System.ComponentModel;
using System.Data;
using System.Drawing;
using System.Linq;
using System.Text;
using System.Windows.Forms; namespace DemoGeometry
{
public partial class MainForm : Form
{
public MainForm()
{
InitializeComponent();
} private void btnLine_Click(object sender, EventArgs e)
{
if (this.linearControl1.IsValid())
{
var a = this.linearControl1.A;
var b = this.linearControl1.B;
var c = this.linearControl1.C;
//判断方程的参数，是否有效
LinearEquation linear = new LinearEquation() { A = a, B = b, C = c };
if (!linear.IsValid())
{
MessageBox.Show("输入的方程参数无效");
return;
}
if (!this.axisControl1.CheckLineIsValid(linear))
{
MessageBox.Show("输入的方程不在坐标轴内");
return;
}
bool flag = this.axisControl1.GenerateLinear(linear);
if (!flag)
{
MessageBox.Show("生成直线失败");
return;
}
}
} private void btnClear_Click(object sender, EventArgs e)
{
this.axisControl1.Clear();
} private void btnPoint_Click(object sender, EventArgs e)
{
if (this.pointControl1.IsValid())
{
float x = this.pointControl1.X;
float y = this.pointControl1.Y;
PointF point = new PointF(x, y);
if (!this.axisControl1.CheckPointIsValid(point)) {
MessageBox.Show("输入的点不在坐标轴内");
return;
}
bool flag = this.axisControl1.GeneratePoint(point);
if (!flag)
{
MessageBox.Show("生成点失败");
return;
}
} } private void btnParabolic_Click(object sender, EventArgs e)
{
if (this.parabolicControl1.IsValid())
{
var a = this.parabolicControl1.A;
var b = this.parabolicControl1.B;
var c = this.parabolicControl1.C;
//判断方程的参数，是否有效
ParabolicEquation parabolic = new ParabolicEquation() { A = a, B = b, C = c };
if (!parabolic.IsValid())
{
MessageBox.Show("输入的方程参数无效");
return;
}
if (!this.axisControl1.CheckParabolicIsValid(parabolic))
{
MessageBox.Show("输入的方程不在坐标轴内");
return;
}
bool flag = this.axisControl1.GenerateParabolic(parabolic);
if (!flag)
{
MessageBox.Show("生成抛物线失败");
return;
}
}
}
}
}
【扩展多边形代码】
/// <summary>
/// 生成多边形
/// </summary>
/// <param name="lstPoints"></param>
/// <returns></returns>
public bool GeneratePolygon(List<PointF> lstPoints ,bool isOriginal=true) { Color lineColor = Color.Red;//线条的颜色
if (isOriginal)
{
lineColor = Color.Red;
}
else {
lineColor = Color.Blue;
} Graphics g = this.CreateGraphics();
//画点
foreach (var p in lstPoints) {
this.GeneratePoint(p);
}
//将刻度点，转换成像素点
List<PointF> lstPixlsPoint = ConvertScaleToPixls(lstPoints);
//绘制多边形
g.DrawPolygon(new Pen(lineColor,), lstPixlsPoint.ToArray());
return true;
} /// <summary>
/// 扩展多边形，即在现有多边形的基础上进行扩展
/// </summary>
/// <param name="lstPoints"></param>
/// <returns></returns>
public bool GenerateExpandPolygon(List<PointF> lstPoints,PointF center,float distance) {
//1.求多边形对应的外扩平行斜线
List<LinearEquation> lstLines = new List<LinearEquation>();
int len = lstPoints.Count();
for (int i = ; i < len; i++) {
var p0 = lstPoints[i];//第i个元素
var p1 = lstPoints[(i + ) % len];//第i+1个元素
LinearEquation linearEquation = new LinearEquation();
if (p0.X == p1.X)
{
//垂直于x轴，没有斜率
linearEquation.A = ;
linearEquation.B = ;
if (p0.X > center.X)
{
linearEquation.C = -(p0.X + distance);
}
else {
linearEquation.C = -(p0.X - distance);
} }
else if (p0.Y == p1.Y)
{
//垂直于y轴，斜率为0
linearEquation.A = ;
linearEquation.B = ;
if (p0.Y > center.Y)
{
linearEquation.C = -(p0.Y + distance);
}
else {
linearEquation.C = -(p0.Y - distance);
}
}
else {
//先求两点对应的点斜式方程y=kx+b
float k = (p0.Y - p1.Y) / (p0.X - p1.X);
float b = p0.Y - k * p0.X;
//求出平行线对应的b即可。
float b_center = center.Y - k * center.X;
linearEquation.A = k;
linearEquation.B = -;//如果此处为-1,则C=b if (b > b_center)
{
//如果在原点上方 linearEquation.C =(b+(float) Math.Abs((distance/(Math.Sin(-Math.PI/-Math.Atan(k))))));
}
else {
//如果在原点下方
linearEquation.C = (b - (float)Math.Abs((distance / (Math.Sin(-Math.PI / - Math.Atan(k))))));
}
}
//this.GenerateLinear(linearEquation);
lstLines.Add(linearEquation);
}
List<PointF> lstNewPoints = new List<PointF>();
//2.求相邻外扩平行斜线的交点
for (int i = ; i < len; i++) {
var line0 = lstLines[i];
var line1 = lstLines[(i + ) % len];
float x = 0.0f;
float y = 0.0f;
if (line0.A == 0.0f)
{
y = -line0.C / line0.B;
x= line1.GetValueFromY(y);
}
else if (line0.B == 0.0f)
{
x = -line0.C / line0.A;
y = line1.GetValueFromX(x);
}
else if (line1.A == 0.0f)
{
y = -line1.C / line1.B;
x = line0.GetValueFromY(y);
}
else if (line1.B == 0.0f)
{
x = -line1.C / line1.A;
y = line0.GetValueFromX(x);
}
else {
//两个都有斜率的直线的交点
float k0 = -line0.A / line0.B;
float k1 = -line1.A / line1.B;
float b0 = -line0.C / line0.B;
float b1 = -line1.C / line1.B;
x = (b1 - b0) / (k0 - k1);
y = line0.GetValueFromX(x);
}
lstNewPoints.Add(new PointF(x, y));
}
this.GeneratePolygon(lstNewPoints, false);
//
return true;
}
备注：源码 点击下方链接
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