Qt5 使用 QPainter 绘制阿基米德螺线

前言

阿基米德螺线在工控运动控制、CAM 绘图、轨迹仿真场景经常用到。极坐标公式 r=a*θ，需要转换成直角坐标x,y，再用大量微小直线段拟合绘制。 本文使用 Qt5.9/Qt5.15，QPainter实现绘图，效果图如下：

环境：Windows Qt5.15.2 MSVC，同样可以直接交叉编译运行在嵌入式 Linux。

数学原理

阿基米德螺线极坐标：

r=a*θ

• a：螺线扩张系数；a 越大，圈与圈之间间距越大
• θ：极角，单位弧度；2 PI代表完整一圈

转换直角坐标：

注意：阿基米德螺线不是圆弧，不能使用 drawArc 圆弧 API，只能大量短直线逐段拟合。运动控制里也是输出连续微小线段给控制器做直线插补。

完整源码

widget.h

#ifndef WIDGET_H
#define WIDGET_H

#include <QWidget>
namespace Ui {
class Widget;
}

class Widget : public QWidget
{
Q_OBJECT
public:
explicit Widget(QWidget *parent = nullptr);
~Widget();
protected:
void paintEvent(QPaintEvent *event) override;

private:
Ui::Widget *ui;
};

#endif // WIDGET_H

widget.cpp

#include "widget.h"
#include "ui_widget.h"
#include <QPainter>
#include <QPaintEvent>
#include <cmath>

Widget::Widget(QWidget *parent) :
QWidget(parent),
ui(new Ui::Widget)
{
ui->setupUi(this);
resize(600,600);
}

Widget::~Widget()
{
delete ui;
}

void Widget::paintEvent(QPaintEvent *event)
{
Q_UNUSED(event);
QPainter painter(this);
//开启抗锯齿，曲线更顺滑
painter.setRenderHint(QPainter::Antialiasing);

//坐标系平移：把绘图原点移动到窗口中心
int cx = width() / 2;
int cy = height() / 2;
painter.translate(cx, cy);
painter.scale(1, -1); // Y轴翻转，Y向上，变成教科书逆时针阿基米德螺线

// ========= 绘制中心十字线 =========
int crossLen = 120; // 十字半长，向上下左右各延伸120像素，调大就更长
painter.setPen(QPen(Qt::red,1)); //红色细线画十字
painter.drawLine(-crossLen, 0, crossLen, 0); //水平横线
painter.drawLine(0, -crossLen, 0, crossLen); //垂直竖线

//画笔：蓝色，线宽2像素，切回螺线画笔
painter.setPen(QPen(Qt::blue,2));

double a = 4.0; //螺线疏密系数，越大圈越松散
double thetaMax = 8*M_PI; //总角度，6π =3圈
double step = 0.03; //角度步长，越小曲线越平滑

QPointF prevPt;
bool first = true;

//for(double theta = thetaMax; theta >= 0; theta -= step)//由外向内
for(double theta = 0; theta <= thetaMax; theta += step)//由内向外
{
double r = a * theta;
double x = r * std::cos(theta);
double y = r * std::sin(theta);
QPointF pt(x, y);
if(!first)
{
//前后两点绘制微小直线，拟合螺线
painter.drawLine(prevPt, pt);
}
prevPt = pt;
first = false;
}
}

main.cpp

#include <QApplication>
#include "widget.h"

int main(int argc, char *argv[])
{
QApplication app(argc, argv);
Widget w;
w.show();
return app.exec();
}

.pro 工程配置

QT += core gui widgets

TARGET = ArchimedeanSpiralDemo
TEMPLATE = app

SOURCES += main.cpp\\
widget.cpp

HEADERS += widget.h

FORMS += widget.ui

编译踩坑点

#define _USE_MATH_DEFINES
#include <cmath>

#include <QPainter>
#include <QPaintEvent>

参数调参说明

表格

示例：a=2.0; thetaMax=12*M_PI; step=0.02，可以生成 6 圈更加紧密的螺线。

如果使用mfc gdi

#include <cmath>

#include <gdiplus.h>
#pragma comment(lib,"gdiplus.lib")
using namespace Gdiplus;
class CGdiplusInit
{
public:
CGdiplusInit()
{
GdiplusStartupInput gdiInput;
GdiplusStartup(&m_token, &gdiInput, NULL);
}
~CGdiplusInit()
{
GdiplusShutdown(m_token);
}
private:
ULONG_PTR m_token;
};
static CGdiplusInit g_gdiplusInit;

CRect rc;
GetClientRect(&rc);
int cx = rc.Width() / 2;
int cy = rc.Height() / 2;
Graphics g(dc.m_hDC);
g.SetSmoothingMode(SmoothingModeAntiAlias);
Matrix mat;
mat.Translate((REAL)cx, (REAL)cy);
mat.Scale(1.0f, -1.0f);
g.SetTransform(&mat);

// ========= 红色十字线 =========
int crossLen = 120;
Pen penRed(Color(255, 255, 0, 0), 1.0f);
PointF crossPts[] = {
PointF((REAL)-crossLen, 0),
PointF((REAL)crossLen, 0),
PointF(0, (REAL)-crossLen),
PointF(0, (REAL)crossLen)
};
g.DrawLine(&penRed, crossPts[0], crossPts[1]);
g.DrawLine(&penRed, crossPts[2], crossPts[3]);

// ========= 蓝色阿基米德螺线 =========
Pen penBlue(Color(255, 0, 0, 255), 2.0f);
const double PI = 3.14159265358979323846;
double a = 4.0;
double thetaMax = 8 * PI;
double step = 0.03;

// 扩大数组预留空间，+10，防止浮点误差导致点数不足
int pointCount = static_cast<int>((thetaMax / step) + 10);
PointF* pts = new PointF[pointCount];
int idx = 0;

double theta;
for (theta = 0.0; theta < thetaMax; theta += step)
{
double r = a * theta;
double x = r * cos(theta);
double y = r * sin(theta);
pts[idx].X = static_cast<REAL>(x);
pts[idx].Y = static_cast<REAL>(y);
idx++;
}
//强制补上最后一个点：thetaMax，把缺失的尾巴补齐
{
double r = a * thetaMax;
double x = r * cos(thetaMax);
double y = r * sin(thetaMax);
pts[idx].X = static_cast<REAL>(x);
pts[idx].Y = static_cast<REAL>(y);
idx++;
}

g.DrawLines(&penBlue, pts, idx);
delete[] pts;