1、贝塞尔曲线

• 贝塞尔曲线（Bézier curve），又称贝兹曲线或贝济埃曲线，是应用于二维图形应用程序的数学曲线。一般的矢量图形软件通过它来精确画出曲线，贝兹曲线由线段与节点组成，节点是可拖动的支点，线段像可伸缩的皮筋，我们在绘图工具上看到的钢笔工具就是来做这种矢量曲线的。贝塞尔曲线是计算机图形学中相当重要的参数曲线，在一些比较成熟的位图软件中也有贝塞尔曲线工具，如 PhotoShop 等。在 Flash4 中还没有完整的曲线工具，而在 Flash5 里面已经提供出贝塞尔曲线工具。

• 二阶贝塞尔曲线示意图

• 三阶贝塞尔曲线示意图

• 贝塞尔路径（UIBezierPath）是 iOS UIKit 框架中对 Quartz 2D 绘图的封装。实际操作起来，使用贝塞尔路径，更为方便。用法与 CGContextRef 类似，但是 OC 对其进行了封装，更加面向对象。

• 贝塞尔路径常用的方法

      // 设置起始点
      - (void)moveToPoint:(CGPoint)point;
  
      // 添加直线到一点
      - (void)addLineToPoint:(CGPoint)point;
  
      // 封闭闭路径
      - (void)closePath;
  
      // 返回一个描述椭圆的路径
      + (UIBezierPath *)bezierPathWithOvalInRect:(CGRect)rect;
  
      // 贝塞尔曲线
      - (void)addQuadCurveToPoint:(CGPoint)endPoint controlPoint:(CGPoint)controlPoint;
  
      // 三次贝塞尔曲线
      - (void)addCurveToPoint:(CGPoint)endPoint controlPoint1:(CGPoint)controlPoint1 controlPoint2:(CGPoint)controlPoint2;
  
      // 绘制圆弧
      - (void)addArcWithCenter:(CGPoint)center radius:(CGFloat)radius startAngle:(CGFloat)startAngle endAngle:(CGFloat)endAngle clockwise:(BOOL)clockwise;

2、基本图形的绘制

• 贝塞尔路径基本图形的绘制详情见 Quartz 2D 二维绘图 基本图形绘制

3、二三阶贝塞尔曲线示例

3.1 二阶贝塞尔曲线示例

• QBezierPathView.h

      @interface QBezierPathView : UIView
  
      @end

• QBezierPathView.m

      @interface QBezierPathView ()
  
      /// 路径
      @property (nonatomic, strong) UIBezierPath *path;
  
      /// 起始点
      @property (nonatomic, assign) CGPoint startP;
  
      /// 终止点
      @property (nonatomic, assign) CGPoint endP;
  
      /// 控制点
      @property (nonatomic, assign) CGPoint controlP;
  
      /// 线的颜色
      @property (nonatomic, strong) UIColor *pathColor;
  
      /// 线的宽度
      @property (nonatomic, assign) CGFloat pathWidth;
  
      /// 当前触摸的点
      @property (nonatomic, assign) NSUInteger currentTouchP;
  
      @end
  
      @implementation QBezierPathView
  
      /// 初始化
      - (instancetype)initWithFrame:(CGRect)frame {
  
          if (self = [super initWithFrame:frame]) {
  
              // 设置初始值
  
              self.startP    = CGPointMake(20, 300);
              self.endP      = CGPointMake(250, 300);
              self.controlP  = CGPointMake(100, 100);
  
              self.pathColor = [UIColor redColor];
              self.pathWidth = 2;
          }
          return self;
      }
  
      /// 绘制二阶贝塞尔曲线
      - (void)drawRect:(CGRect)rect {
  
          // 绘制贝塞尔曲线
  
          self.path = [UIBezierPath bezierPath];
          [self.path moveToPoint:self.startP];
  
          [self.path addQuadCurveToPoint:self.endP controlPoint:self.controlP];
  
          self.path.lineWidth = self.pathWidth;
          [self.pathColor setStroke];
          [self.path stroke];
  
          // 绘制辅助线
  
          self.path = [UIBezierPath bezierPath];
  
          self.path.lineWidth = 1;
          [[UIColor grayColor] setStroke];
  
          CGFloat lengths[] = {5};
          [self.path setLineDash:lengths count:1 phase:1];
  
          [self.path moveToPoint:self.controlP];
          [self.path addLineToPoint:self.startP];
          [self.path stroke];
  
          [self.path moveToPoint:self.controlP];
          [self.path addLineToPoint:self.endP];
          [self.path stroke];
  
          // 绘制辅助点及信息
  
          self.path = [UIBezierPath bezierPathWithArcCenter:self.startP radius:4 startAngle:0 endAngle:M_PI * 2 clockwise:YES];
          [[UIColor blackColor] setStroke];
          [self.path fill];
  
          self.path = [UIBezierPath bezierPathWithArcCenter:self.endP radius:4 startAngle:0 endAngle:M_PI * 2 clockwise:YES];
          [[UIColor blackColor] setStroke];
          [self.path fill];
  
          self.path = [UIBezierPath bezierPathWithArcCenter:self.controlP radius:3 startAngle:0 endAngle:M_PI * 2 clockwise:YES];
          self.path.lineWidth = 2;
          [[UIColor blackColor] setStroke];
          [self.path stroke];
  
          CGRect startMsgRect = CGRectMake(self.startP.x + 8, self.startP.y - 7, 50, 20);
          [@"起始点" drawInRect:startMsgRect withAttributes:nil];
  
          CGRect endMsgRect = CGRectMake(self.endP.x + 8, self.endP.y - 7, 50, 20);
          [@"终止点" drawInRect:endMsgRect withAttributes:nil];
  
          CGRect control1MsgRect = CGRectMake(self.controlP.x + 8, self.controlP.y - 7, 50, 20);
          [@"控制点" drawInRect:control1MsgRect withAttributes:nil];
      }
  
      /// 触摸开始
      - (void)touchesBegan:(NSSet<UITouch *> *)touches withEvent:(nullable UIEvent *)event {
  
          // 获取触摸点位置
          CGPoint startPoint = [touches.anyObject locationInView:self];
  
          CGRect startR   = CGRectMake(self.startP.x   - 4, self.startP.y   - 4, 8, 8);
          CGRect endR     = CGRectMake(self.endP.x     - 4, self.endP.y     - 4, 8, 8);
          CGRect controlR = CGRectMake(self.controlP.x - 4, self.controlP.y - 4, 8, 8);
  
          // 判断当前触摸点
          if (CGRectContainsPoint(startR, startPoint)) {
              self.currentTouchP = 1;
          } else if (CGRectContainsPoint(endR, startPoint)) {
              self.currentTouchP = 2;
          } else if (CGRectContainsPoint(controlR, startPoint)) {
              self.currentTouchP = 3;
          }
      }
  
      /// 触摸移动
      - (void)touchesMoved:(NSSet<UITouch *> *)touches withEvent:(nullable UIEvent *)event {
  
          // 获取触摸点位置
          CGPoint touchPoint = [touches.anyObject locationInView:self];
  
          // 限制触摸点的边界
  
          if (touchPoint.x < 0) {
              touchPoint.x = 0;
          }
  
          if (touchPoint.x > self.bounds.size.width) {
              touchPoint.x = self.bounds.size.width;
          }
  
          if (touchPoint.y < 0) {
              touchPoint.y = 0;
          }
  
          if (touchPoint.y > self.bounds.size.height) {
              touchPoint.y = self.bounds.size.height;
          }
  
          // 设置当前触摸点的值
          switch (self.currentTouchP) {
              case 1:
                  self.startP = touchPoint;
                  break;
  
              case 2:
                  self.endP = touchPoint;
                  break;
  
              case 3:
                  self.controlP = touchPoint;
                  break;
  
              default:
                  break;
          }
  
          // 刷新
          [self setNeedsDisplay];
      }
  
      /// 触摸结束
      - (void)touchesEnded:(NSSet<UITouch *> *)touches withEvent:(nullable UIEvent *)event {
  
          // 释放之前的触摸点
          self.currentTouchP = 0;
      }
  
      /// 触摸取消
      - (void)touchesCancelled:(NSSet *)touches withEvent:(nullable UIEvent *)event {
          [self touchesEnded:touches withEvent:event];
      }
  
      @end

• ViewController.m

      CGRect frame = CGRectMake(20, 50, self.view.bounds.size.width - 40, 400);
  
      QBezierPathView *pathView = [[QBezierPathView alloc] initWithFrame:frame];
  
      pathView.backgroundColor = [UIColor whiteColor];
      pathView.layer.borderWidth = 1;
  
      [self.view addSubview:pathView];

• 效果

3.2 三阶贝塞尔曲线示例

• QBezierPathView.h

      @interface QBezierPathView : UIView
  
      @end

• QBezierPathView.m

      @interface QBezierPathView : UIView
      @interface QBezierPathView ()
  
      /// 路径
      @property (nonatomic, strong) UIBezierPath *path;
  
      /// 起始点
      @property (nonatomic, assign) CGPoint startP;
  
      /// 终止点
      @property (nonatomic, assign) CGPoint endP;
  
      /// 控制点
      @property (nonatomic, assign) CGPoint controlP1;
      @property (nonatomic, assign) CGPoint controlP2;
  
      /// 线的颜色
      @property (nonatomic, strong) UIColor *pathColor;
  
      /// 线的宽度
      @property (nonatomic, assign) CGFloat pathWidth;
  
      /// 当前触摸的点
      @property (nonatomic, assign) NSUInteger currentTouchP;
  
      @end
  
      @implementation QBezierPathView
  
      /// 初始化
      - (instancetype)initWithFrame:(CGRect)frame {
  
          if (self = [super initWithFrame:frame]) {
  
              // 设置初始值
  
              self.startP    = CGPointMake(20, 300);
              self.endP      = CGPointMake(250, 300);
              self.controlP1  = CGPointMake(100, 100);
              self.controlP2  = CGPointMake(200, 350);
  
              self.pathColor = [UIColor redColor];
              self.pathWidth = 2;
          }
          return self;
      }
  
      /// 绘制二阶贝塞尔曲线
      - (void)drawRect:(CGRect)rect {
  
          // 绘制贝塞尔曲线
  
          self.path = [UIBezierPath bezierPath];
          [self.path moveToPoint:self.startP];
  
          [self.path addCurveToPoint:self.endP controlPoint1:self.controlP1 controlPoint2:self.controlP2];
  
          self.path.lineWidth = self.pathWidth;
          [self.pathColor setStroke];
          [self.path stroke];
  
          // 绘制辅助线
  
          self.path = [UIBezierPath bezierPath];
  
          self.path.lineWidth = 1;
          [[UIColor grayColor] setStroke];
  
          CGFloat lengths[] = {5};
          [self.path setLineDash:lengths count:1 phase:1];
  
          [self.path moveToPoint:self.controlP1];
          [self.path addLineToPoint:self.startP];
          [self.path stroke];
  
          [self.path moveToPoint:self.controlP1];
          [self.path addLineToPoint:self.controlP2];
          [self.path stroke];
  
          [self.path moveToPoint:self.controlP2];
          [self.path addLineToPoint:self.endP];
          [self.path stroke];
  
          // 绘制辅助点及信息
  
          self.path = [UIBezierPath bezierPathWithArcCenter:self.startP radius:4 startAngle:0 endAngle:M_PI * 2 clockwise:YES];
          [[UIColor blackColor] setStroke];
          [self.path fill];
  
          self.path = [UIBezierPath bezierPathWithArcCenter:self.endP radius:4 startAngle:0 endAngle:M_PI * 2 clockwise:YES];
          [[UIColor blackColor] setStroke];
          [self.path fill];
  
          self.path = [UIBezierPath bezierPathWithArcCenter:self.controlP1 radius:3 startAngle:0 endAngle:M_PI * 2 clockwise:YES];
          self.path.lineWidth = 2;
          [[UIColor blackColor] setStroke];
          [self.path stroke];
  
          self.path = [UIBezierPath bezierPathWithArcCenter:self.controlP2 radius:3 startAngle:0 endAngle:M_PI * 2 clockwise:YES];
          self.path.lineWidth = 2;
          [[UIColor blackColor] setStroke];
          [self.path stroke];
  
          CGRect startMsgRect = CGRectMake(self.startP.x + 8, self.startP.y - 7, 50, 20);
          [@"起始点" drawInRect:startMsgRect withAttributes:nil];
  
          CGRect endMsgRect = CGRectMake(self.endP.x + 8, self.endP.y - 7, 50, 20);
          [@"终止点" drawInRect:endMsgRect withAttributes:nil];
  
          CGRect control1MsgRect = CGRectMake(self.controlP1.x + 8, self.controlP1.y - 7, 50, 20);
          [@"控制点1" drawInRect:control1MsgRect withAttributes:nil];
  
          CGRect control2MsgRect = CGRectMake(self.controlP2.x + 8, self.controlP2.y - 7, 50, 20);
          [@"控制点2" drawInRect:control2MsgRect withAttributes:nil];
      }
  
      /// 触摸开始
      - (void)touchesBegan:(NSSet<UITouch *> *)touches withEvent:(nullable UIEvent *)event {
  
          // 获取触摸点位置
          CGPoint startPoint = [touches.anyObject locationInView:self];
  
          CGRect startR    = CGRectMake(self.startP.x   - 4, self.startP.y   - 4, 8, 8);
          CGRect endR      = CGRectMake(self.endP.x     - 4, self.endP.y     - 4, 8, 8);
          CGRect controlR1 = CGRectMake(self.controlP1.x - 4, self.controlP1.y - 4, 8, 8);
          CGRect controlR2 = CGRectMake(self.controlP2.x - 4, self.controlP2.y - 4, 8, 8);
  
          // 判断当前触摸点
          if (CGRectContainsPoint(startR, startPoint)) {
              self.currentTouchP = 1;
          } else if (CGRectContainsPoint(endR, startPoint)) {
              self.currentTouchP = 2;
          } else if (CGRectContainsPoint(controlR1, startPoint)) {
              self.currentTouchP = 3;
          } else if (CGRectContainsPoint(controlR2, startPoint)) {
              self.currentTouchP = 4;
          }
      }
  
      /// 触摸移动
      - (void)touchesMoved:(NSSet<UITouch *> *)touches withEvent:(nullable UIEvent *)event {
  
          // 获取触摸点位置
          CGPoint touchPoint = [touches.anyObject locationInView:self];
  
          // 限制触摸点的边界
  
          if (touchPoint.x < 0) {
              touchPoint.x = 0;
          }
  
          if (touchPoint.x > self.bounds.size.width) {
              touchPoint.x = self.bounds.size.width;
          }
  
          if (touchPoint.y < 0) {
              touchPoint.y = 0;
          }
  
          if (touchPoint.y > self.bounds.size.height) {
              touchPoint.y = self.bounds.size.height;
          }
  
          // 设置当前触摸点的值
          switch (self.currentTouchP) {
              case 1:
                  self.startP = touchPoint;
                  break;
  
              case 2:
                  self.endP = touchPoint;
                  break;
  
              case 3:
                  self.controlP1 = touchPoint;
                  break;
  
              case 4:
                  self.controlP2 = touchPoint;
                  break;
  
              default:
                  break;
          }
  
          // 刷新
          [self setNeedsDisplay];
      }
  
      /// 触摸结束
      - (void)touchesEnded:(NSSet<UITouch *> *)touches withEvent:(nullable UIEvent *)event {
  
          // 释放之前的触摸点
          self.currentTouchP = 0;
      }
  
      /// 触摸取消
      - (void)touchesCancelled:(NSSet *)touches withEvent:(nullable UIEvent *)event {
          [self touchesEnded:touches withEvent:event];
      }
  
      @end

• ViewController.m

      CGRect frame = CGRectMake(20, 50, self.view.bounds.size.width - 40, 400);
  
      QBezierPathView *pathView = [[QBezierPathView alloc] initWithFrame:frame];
  
      pathView.backgroundColor = [UIColor whiteColor];
      pathView.layer.borderWidth = 1;
  
      [self.view addSubview:pathView];

• 效果