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

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1、贝塞尔曲线

2、基本图形的绘制

3、二三阶贝塞尔曲线示例

3.1 二阶贝塞尔曲线示例

3.2 三阶贝塞尔曲线示例

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