如何启发式地制作平行贝塞尔曲线

我只发现这个博客有相关答案http://seant23.wordpress.com/2010/11/12/offset-bezier-curves/，但不幸的是我不懂这种语言，也无法理解其背后的数学。我需要的是知道如何使贝塞尔曲线与我现有的曲线平行。

我有一个点、段和路径类，但我不明白如何将路径分成段。 Point 类具有 CGPoint 位置公共变量， Segment 类具有 4 个点属性，Point *control1、*control2、*point2 和 *point1； Path 类包含一个 NSMutableArray 段和一个 Point startPoint。

我是 Objective C 的新手，如果不是为了我的特定类构建，至少对于更通用的方法，我真的很感激一些帮助。

我不知道您正在解决的具体问题，但一个可爱（而且非常简单）的解决方案是仅渲染贝塞尔曲线的轮廓，例如：

使用 Core Graphics 可以轻松完成此操作（在本例中，drawRect of a UIView子类）：

- (void)drawRect:(CGRect)rect {
    CGPathRef path = [self newBezierPath];
    CGPathRef outlinePath = CGPathCreateCopyByStrokingPath(path, NULL, 10, kCGLineCapButt, kCGLineJoinBevel, 0);

    CGContextRef context = UIGraphicsGetCurrentContext();
    CGContextSetLineWidth(context, 3.0);
    CGContextAddPath(context, outlinePath);
    CGContextSetStrokeColorWithColor(context, [[UIColor redColor] CGColor]);
    CGContextDrawPath(context, kCGPathStroke);

    CGPathRelease(path);
    CGPathRelease(outlinePath);
}

- (CGPathRef)newBezierPath {
    CGPoint point1 = CGPointMake(10.0, 50.0);
    CGPoint point2 = CGPointMake(self.bounds.size.width - 10.0, point1.y + 150.0);

    CGPoint controlPoint1 = CGPointMake(point1.x + 400.0, point1.y);
    CGPoint controlPoint2 = CGPointMake(point2.x - 400.0, point2.y);

    CGMutablePathRef path = CGPathCreateMutable();
    CGPathMoveToPoint(path, NULL, point1.x, point1.y);
    CGPathAddCurveToPoint(path, NULL, controlPoint1.x, controlPoint1.y, controlPoint2.x, controlPoint2.y, point2.x, point2.y);

    return path;
}

或者在 Swift 3 中：

override func draw(_ rect: CGRect) {
    let path = bezierPath().cgPath
    let outlinePath = path.copy(strokingWithWidth: 10, lineCap: .butt, lineJoin: .bevel, miterLimit: 0)

    let context = UIGraphicsGetCurrentContext()!
    context.setLineWidth(3)
    context.addPath(outlinePath)
    context.setStrokeColor(UIColor.red.cgColor)
    context.strokePath()
}

private func bezierPath() -> UIBezierPath {
    let point1 = CGPoint(x: 10.0, y: 50.0)
    let point2 = CGPoint(x: bounds.size.width - 10.0, y: point1.y + 150.0)

    let controlPoint1 = CGPoint(x: point1.x + 400.0, y: point1.y)
    let controlPoint2 = CGPoint(x: point2.x - 400.0, y: point2.y)

    let path = UIBezierPath()
    path.move(to: point1)
    path.addCurve(to: point2, controlPoint1: controlPoint1, controlPoint2: controlPoint2)

    return path
}

如果你真的想画一条平行路径，那就更复杂了。但是您可以渲染类似的东西（原始贝塞尔路径为红色，“平行”线为蓝色）。

我不完全确定您所确定的算法，但我将其呈现为

• 自己计算贝塞尔点（红色路径），将其分解得足够精细以使效果平滑；
• 计算每个点与下一个点之间的角度；
• 通过获取贝塞尔路径上的点来计算偏移路径（蓝色的）的坐标，并计算垂直于我刚刚确定的角度的新点；和
• 使用这些偏移点坐标，绘制一系列新的线段以渲染与贝塞尔曲线的平行线。

因此，在 Objective-C 中，这可能看起来像：

- (void)drawRect:(CGRect)rect {
    CGPoint point1 = CGPointMake(10.0, 50.0);
    CGPoint point2 = CGPointMake(self.bounds.size.width - 10.0, point1.y + 150.0);

    CGPoint controlPoint1 = CGPointMake(point1.x + 400.0, point1.y);
    CGPoint controlPoint2 = CGPointMake(point2.x - 400.0, point2.y);

    // draw original bezier path in red

    [[UIColor redColor] setStroke];

    [[self bezierPathFromPoint1:point1
                         point2:point2
                  controlPoint1:controlPoint1
                  controlPoint2:controlPoint2] stroke];

    // calculate and draw offset bezier curve in blue

    [[UIColor blueColor] setStroke];

    [[self offsetBezierPathBy:10.0
                       point1:point1
                       point2:point2
                controlPoint1:controlPoint1
                controlPoint2:controlPoint2] stroke];
}

- (UIBezierPath *)bezierPathFromPoint1:(CGPoint)point1
                                point2:(CGPoint)point2
                         controlPoint1:(CGPoint)controlPoint1
                         controlPoint2:(CGPoint)controlPoint2 {
    UIBezierPath *path = [UIBezierPath bezierPath];

    [path moveToPoint:point1];
    [path addCurveToPoint:point2 controlPoint1:controlPoint1 controlPoint2:controlPoint2];

    return path;
}

- (UIBezierPath *)offsetBezierPathBy:(CGFloat)offset
                              point1:(CGPoint)point1
                              point2:(CGPoint)point2
                       controlPoint1:(CGPoint)controlPoint1
                       controlPoint2:(CGPoint)controlPoint2 {
    UIBezierPath *path = [UIBezierPath bezierPath];
    static NSInteger numberOfPoints = 100;

    CGPoint previousPoint = [self cubicBezierAtTime:0.0
                                             point1:point1
                                             point2:point2
                                      controlPoint1:controlPoint1
                                      controlPoint2:controlPoint2];
    CGPoint point;
    double angle;

    for (NSInteger i = 1; i <= numberOfPoints; i++) {
        double t = (double) i / numberOfPoints;

        point = [self cubicBezierAtTime:t
                                 point1:point1
                                 point2:point2
                          controlPoint1:controlPoint1
                          controlPoint2:controlPoint2];

        // calculate the angle to the offset point
        // this is the angle between the two points, plus 90 degrees (pi / 2.0)

        angle = atan2(point.y - previousPoint.y, point.x - previousPoint.x) + M_PI_2;


        if (i == 1)
            [path moveToPoint:[self offsetPoint:previousPoint by:offset angle:angle]];

        previousPoint = point;
        [path addLineToPoint:[self offsetPoint:previousPoint by:offset angle:angle]];
    }

    return path;
}

// return point offset by particular distance and particular angle

- (CGPoint)offsetPoint:(CGPoint)point by:(CGFloat)offset angle:(double)angle {
    return CGPointMake(point.x + cos(angle) * offset, point.y + sin(angle) * offset);
}

// Manually calculate cubic bezier curve
//
// B(t) = (1-t)^3 * point1 + 3 * (1-t)^2 * t controlPoint1 + 3 * (1-t) * t^2 * pointPoint2 + t^3 * point2

- (CGPoint)cubicBezierAtTime:(double)t
                      point1:(CGPoint)point1
                      point2:(CGPoint)point2
               controlPoint1:(CGPoint)controlPoint1
               controlPoint2:(CGPoint)controlPoint2 {
    double oneMinusT = 1.0 - t;
    double oneMinusTSquared = oneMinusT * oneMinusT;
    double oneMinusTCubed = oneMinusTSquared * oneMinusT;

    double tSquared = t * t;
    double tCubed = tSquared * t;

    CGFloat x = point1.x * oneMinusTCubed +
    3.0 * oneMinusTSquared * t * controlPoint1.x +
    3.0 * oneMinusT * tSquared * controlPoint2.x +
    tCubed * point2.x;
    CGFloat y = point1.y * oneMinusTCubed +
    3.0 * oneMinusTSquared * t * controlPoint1.y +
    3.0 * oneMinusT * tSquared * controlPoint2.y +
    tCubed * point2.y;

    return CGPointMake(x, y);
}

或者，在 Swift 3 中：

override func draw(_ rect: CGRect) {
    let point1 = CGPoint(x: 10.0, y: 50.0)
    let point2 = CGPoint(x: bounds.size.width - 10.0, y: point1.y + 150.0)

    let controlPoint1 = CGPoint(x: point1.x + 400.0, y: point1.y)
    let controlPoint2 = CGPoint(x: point2.x - 400.0, y: point2.y)

    UIColor.red.setStroke()
    bezierPath(from: point1, to: point2, withControl: controlPoint1, and: controlPoint2).stroke()

    UIColor.blue.setStroke()
    offSetBezierPath(by: 5, from: point1, to: point2, withControl: controlPoint1, and: controlPoint2).stroke()
}

private func bezierPath(from point1: CGPoint, to point2: CGPoint, withControl controlPoint1: CGPoint, and controlPoint2:CGPoint) -> UIBezierPath {
    let path = UIBezierPath()

    path.move(to: point1)
    path.addCurve(to: point2, controlPoint1: controlPoint1, controlPoint2: controlPoint2)

    return path
}

private func offSetBezierPath(by offset: CGFloat, from point1: CGPoint, to point2: CGPoint, withControl controlPoint1: CGPoint, and controlPoint2:CGPoint) -> UIBezierPath {
    let path = UIBezierPath()

    let numberOfPoints = 100

    var previousPoint = cubicBezier(at: 0, point1: point1, point2: point2, controlPoint1: controlPoint1, controlPoint2: controlPoint2)

    for i in 1 ... numberOfPoints {
        let time = CGFloat(i) / CGFloat(numberOfPoints)
        let point = cubicBezier(at: time, point1: point1, point2: point2, controlPoint1: controlPoint1, controlPoint2: controlPoint2)

        // calculate the angle to the offset point
        // this is the angle between the two points, plus 90 degrees (pi / 2.0)

        let angle = atan2(point.y - previousPoint.y, point.x - previousPoint.x) + .pi / 2;

        if i == 1 {
            path.move(to: calculateOffset(of: previousPoint, by: offset, angle: angle))
        }

        previousPoint = point
        path.addLine(to: calculateOffset(of: previousPoint, by: offset, angle: angle))
    }

    return path
}

/// Return point offset by particular distance and particular angle
///
/// - Parameters:
///   - point: Point to offset.
///   - offset: How much to offset by.
///   - angle: At what angle.
///
/// - Returns: New `CGPoint`.

private func calculateOffset(of point: CGPoint, by offset: CGFloat, angle: CGFloat) -> CGPoint {
    return CGPoint(x: point.x + cos(angle) * offset, y: point.y + sin(angle) * offset)
}

/// Manually calculate cubic bezier curve
///
/// B(t) = (1-t)^3 * point1 + 3 * (1-t)^2 * t controlPoint1 + 3 * (1-t) * t^2 * pointPoint2 + t^3 * point2
///
/// - Parameters:
///   - time:          Time, a value between zero and one.
///   - point1:        Starting point.
///   - point2:        Ending point.
///   - controlPoint1: First control point.
///   - controlPoint2: Second control point.
///
/// - Returns: Point on bezier curve.

private func cubicBezier(at time: CGFloat, point1: CGPoint, point2: CGPoint, controlPoint1: CGPoint, controlPoint2: CGPoint) -> CGPoint {
    let oneMinusT = 1.0 - time
    let oneMinusTSquared = oneMinusT * oneMinusT
    let oneMinusTCubed = oneMinusTSquared * oneMinusT

    let tSquared = time * time
    let tCubed = tSquared * time

    var x = point1.x * oneMinusTCubed
    x += 3.0 * oneMinusTSquared * time * controlPoint1.x
    x += 3.0 * oneMinusT * tSquared * controlPoint2.x
    x += tCubed * point2.x

    var y = point1.y * oneMinusTCubed
    y += 3.0 * oneMinusTSquared * time * controlPoint1.y
    y += 3.0 * oneMinusT * tSquared * controlPoint2.y
    y += tCubed * point2.y

    return CGPoint(x: x, y: y)
}