如何启发式制作平行贝塞尔曲线

Question

我只在这个博客中找到了相关答案http://seant23.wordpress.com/2010/11/12/offset-bezier-curves/ ，但是很遗憾，我不懂该语言，也无法理解其背后的数学原理它。 我需要的是知道如何使贝塞尔曲线平行于我的贝塞尔曲线。

我有一个Point，Segment和Path类，但是我不明白如何将路径分成多个段。 Point类具有CGPoint位置公共变量，Segment类具有作为属性的4个点，即Point * control1，* control2，* point2和* point1； Path类包含一个NSMutableArray段和一个Point startPoint。

我对目标c还是陌生的，如果不是针对我的特定类的构造，至少对于更通用的方法，我将不胜感激。

Answer 1

我不知道您要解决的特定问题，但是一种可爱（且非常简单）的解决方案是仅绘制贝塞尔曲线的轮廓，例如：

使用Core Graphics（在这种情况下，是UIView子类的drawRect ）可以轻松完成此操作：

- (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
}

如果您真的想画一条平行的路径，那就更复杂了。 但是，您可以渲染类似的内容（红色的原始贝塞尔曲线路径，蓝色的“平行”线）。

我不确定您确定的算法，但是我通过

• 自己计算贝塞尔曲线点（红色路径），将其细化到足以使效果平滑的程度；
• 计算每个点和下一个点之间的角度；
• 通过获取贝塞尔曲线路径上的点，并计算与刚确定的角度I垂直的新点，来计算偏移路径（蓝色坐标）的坐标； 和
• 使用这些偏移点坐标，绘制一系列新的线段，以将平行线渲染到贝塞尔曲线。

因此，在Objective-C中，它可能类似于：

- (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)
}

Answer 2

您可能会在Sean的博客上看到我的文章的链接，如果没有： http : //pomax.github.io/bezierinfo/#offsetting详细介绍了偏移曲线。 它指的是本文上半部分涉及的一些主题，例如在拐点处分割曲线，但要注意的是：

1. 您不能为任意曲线B创建真正的偏移曲线C。本文解释了原因，但tl; dr是贝塞尔曲线是整数多项式，并且除了在微小边缘情况下，它们的偏移曲线可以用数学函数，但是它不是整数多项式，因此也不是另一个贝塞尔曲线。
2. 您可以展平以弯曲成“看起来像曲线”多边形，然后将其偏移就可以了。 这可以非常快地完成，如果您使用足够的段，看起来就很好。 它只是无法缩放，您必须根据偏移距离确定要使用多少段。
3. 您可以将曲线切成小段，这些段可以偏移而不会因其他曲线而损失精度。 这比较慢，但允许曲线与其偏移之间有任意距离。

如果您的代码库中没有预构建的偏移功能，那么您将必须自己实现它，在这种情况下，您将需要花一两天的时间来开发它，并专门研究它的工作原理（为此，我非常建议您遍历bezier文章。编写偏移算法需要具有依赖于可用功能的功能）。