计算给定水平线处的形状宽度

Question

假设我有这样一个多边形

public partial class Window2 : Window
{
    public Window2()
    {
        InitializeComponent();

        var myPolygon = new Polygon();
        myPolygon.Stroke = Brushes.Black;
        myPolygon.Fill = Brushes.LightSeaGreen;
        myPolygon.StrokeThickness = 2;

        myPolygon.Points = new PointCollection(new Point[] {
            new Point(50,50),
            new Point(50,165),
            new Point(140,165),
            new Point(140,120),
            new Point(70,120),
            new Point(80,70),
            new Point(140,70),
            new Point(140,50)
        });

        this.Content = myPolygon;
    }
}

假设我想绘制从一边到另一边穿过多边形的红线，如下图所示：

我只知道这条线应该站在哪个垂直位置，但是我怎么知道我应该从哪个水平点开始这条线，从哪个水平点结束这条线呢？

我的主要目标是知道这条线在哪个水平点开始，在哪个水平点结束，以便在这条线上排列文本。

如果线在多个地方穿过形状（如下图），我想得到所有线的数组：

请注意，形状可以由直线和拱形组成。

以下是 Adobe Illustrator 排列文本的方式：

如何在 C# 中执行此操作？

谢谢！

注意：如需悬赏，请附上 C# 示例。

Answer 1

注意：此答案不是关于计算适当的线大小（算术结果），而是关于仅在多边形上显示线（视觉结果）。如果您需要数学，请更改问题上的标签。

您可以绘制完整尺寸的线条，并使用与多边形相等的几何形状对其进行剪裁。假设您将多边形和线托管在名为 grid1:

的网格中

private void DrawLine(Polygon myPolygon, int linePos)
{

    var clip = new StreamGeometry();
    using (var context = clip.Open())
    {
        context.BeginFigure(myPolygon.Points.First(), true, true);
        context.PolyLineTo(myPolygon.Points.Skip(1).ToList(), true, false);
    }
    var line = new Line()
    {
        X1 = 0,
        X2 = Width,
        Y1 = linePos,
        Y2 = linePos,
        Stroke = Brushes.Red,
        StrokeThickness = 2,
        Clip = clip
    };

    grid1.Children.Add(line);
}

结合问题中的代码：

var myPolygon = new Polygon();
myPolygon.Stroke = Brushes.Black;
myPolygon.Fill = Brushes.LightSeaGreen;
myPolygon.StrokeThickness = 2;

myPolygon.Points = new PointCollection(new Point[]
{
    new Point(50,50),
    new Point(50,165),
    new Point(140,165),
    new Point(140,120),
    new Point(70,120),
    new Point(80,70),
    new Point(140,70),
    new Point(140,50)
});

grid1.Children.Add(myPolygon);

DrawLine(myPolygon, 80);
DrawLine(myPolygon, 150);

感谢 Clemens WPF Clipping with a shape 从点创建几何的方法。

如果您计划绘制多条线，您可以选择在父面板上定义一个剪裁，这样内部的所有内容都将被剪裁到多边形边界。

Answer 2

我会说你唯一的选择是测试红线是否与多边形的 any/some segments/arcs 相交。

蛮力是第一个想法。第二个想法是使用 Fortune's algorithm

因为你的红线是水平的，这有点帮助。

如果将多边形的片段存储在按 Y 最小坐标排序的数组中，则可以从测试中跳过红线以下的所有片段。
这个下限很容易通过二分查找找到。

一些实用链接：
How do you detect where two line segments intersect?
Line Segment Circle Intersection

Answer 3

你要把形状分成直线和曲线
检查下面提供的代码，了解这些与您的红线相交的直线/曲线
最后你会得到至少两条相交的直线/曲线，这样你就会知道红线的宽度。

检查线相交的代码：

    public static Point GetLineLineIntersections(
        Point start1, Point end1,
        Point start2, Point end2)
    {
        return GetLineLineIntersections(start1.X, start1.Y,
            end1.X, end1.Y,
            start2.X, start2.Y,
            end2.X, end2.Y);
    }

    public static Point GetLineLineIntersections(
        double x1, double y1,
        double x2, double y2,
        double x3, double y3,
        double x4, double y4)
    {
        double px = ((x1 * y2 - y1 * x2) * (x3 - x4) - (x1 - x2) * (x3 * y4 - y3 * x4)) /
            ((x1 - x2) * (y3 - y4) - (y1 - y2) * (x3 - x4));

        double py = ((x1 * y2 - y1 * x2) * (y3 - y4) - (y1 - y2) * (x3 * y4 - y3 * x4)) /
            ((x1 - x2) * (y3 - y4) - (y1 - y2) * (x3 - x4));

        return new Point(px, py);
    }

检查线与曲线相交的代码：

    public static List<Point> GetLineCurveIntersections(
               Point curve1, Point curve2, Point curve3, Point curve4,
               Point lineStart, Point lineEnd)
    {
        var res = new List<Point>();

        var points = new List<Point>(new Point[] { curve1, curve2, curve3, curve4 });
        Rect rect = pointsBoundingRect(points);
        var rectData = new Tuple<Rect, List<Point>>(rect, points);
        var rectsData = new Queue<Tuple<Rect, List<Point>>>();
        rectsData.Enqueue(rectData);

        while (rectsData.Count != 0)
        {
            rectData = rectsData.Dequeue();
            rect = rectData.Item1;
            var controlPoints = rectData.Item2;
            if (!lineIntersectsRect(lineStart, lineEnd, rect))
                continue;

            if (isRectSmallEnough(rect))
            {
                res.Add(rect.Location);
                continue;
            }

            var pointsLeft = controlPointsForCurveInRange(0, 0.5, controlPoints);
            var pointsRight = controlPointsForCurveInRange(0.501, 1, controlPoints);
            var rectLeft = pointsBoundingRect(pointsLeft);
            var rectRight = pointsBoundingRect(pointsRight);

            rectsData.Enqueue(new Tuple<Rect, List<Point>>(rectLeft, pointsLeft));
            rectsData.Enqueue(new Tuple<Rect, List<Point>>(rectRight, pointsRight));
        }

        return res;
    }

    static Rect pointsBoundingRect(List<Point> points)
    {
        var xMin = points[0].X;
        var yMin = points[0].Y;
        var xMax = xMin;
        var yMax = yMin;

        for (var i = 0; i < points.Count; ++i)
        {
            var x = points[i].X;
            var y = points[i].Y;
            if (x < xMin)
                xMin = x;
            if (x > xMax)
                xMax = x;
            if (y < yMin)
                yMin = y;
            if (y > yMax)
                yMax = y;
        }

        return new Rect(new Point(xMax, yMax), new Point(xMin, yMin));
    }

    static bool lineIntersectsRect(Point lineStart, Point lineEnd, Rect rect)
    {
        var lineXmin = lineStart.X;
        var lineXmax = lineEnd.X;

        if (lineXmin > lineXmax)
        {
            lineXmin = lineEnd.X;
            lineXmax = lineStart.X;
        }

        if (lineXmax > rect.BottomRight.X)
            lineXmax = rect.BottomRight.X;

        if (lineXmin < rect.Location.X)
            lineXmin = rect.Location.X;

        if (lineXmin > lineXmax)
            return false;

        var minY = lineStart.Y;
        var maxY = lineEnd.Y;

        var dx = lineEnd.X - lineStart.X;
        if (Math.Abs(dx) > 0.000001)
        {
            //line equation
            var a = (lineEnd.Y - lineStart.Y) / dx;
            var b = lineStart.Y - a * lineStart.X;
            minY = a * lineXmin + b;
            maxY = a * lineXmax + b;
        }

        if (minY > maxY)
        {
            var tmp = minY;
            minY = maxY;
            maxY = tmp;
        }

        if (maxY > rect.BottomRight.Y)
            maxY = rect.BottomRight.Y;

        if (minY < rect.Location.Y)
            minY = rect.Location.Y;

        if (minY > maxY)
            return false;

        return true;
    }

    static bool isRectSmallEnough(Rect rect)
    {
        return rect.Width * rect.Height <= 1;
    }

    static Point calculatePointForParameters(double[] parameters, List<Point> controlPoints)
    {
        //De Casteljau's algorithm

        if (parameters.Length != (controlPoints.Count - 1))
        {
            throw new Exception("Invalid input(calculate curve point)");
        }

        if (controlPoints.Count == 1)
            return controlPoints[0];

        var points = controlPoints;
        var iteration = 0;
        while (points.Count != 1)
        {
            var t = parameters[iteration];
            var newPoints = new List<Point>();
            for (var i = 1; i < points.Count; ++i)
            {
                var x = (1 - t) * points[i - 1].X + t * points[i].X;
                var y = (1 - t) * points[i - 1].Y + t * points[i].Y;

                newPoints.Add(new Point(x, y));
            }

            ++iteration;
            points = newPoints;
        }

        return points[0];
    }

    static List<Point> controlPointsForCurveInRange(double tMin, double tMax, List<Point> points)
    {
        var controlPoints = new List<Point>();
        var pointsCount = points.Count;

        var parameters = new double[pointsCount - 1];
        for (var i = 0; i < pointsCount; ++i)
        {
            parameters.Fill(tMin, 0, parameters.Length - i);
            parameters.Fill(tMax, parameters.Length - i, pointsCount);
            var newPoint = calculatePointForParameters(parameters, points);
            controlPoints.Add(newPoint);
        }

        return controlPoints;
    }

public static class Ex
{
    public static void Fill<T>(this IList<T> list, T value, int start, int end)
    {
        end = Math.Min(list.Count, end);
        for (int i = start; i < end; ++i)
        {
            list[i] = value;
        }
    }
}

Answer 4

WPF内置了很多算法，可以避免为像我这样的懒人编写复杂的算法。如果使用得当，Geometry class 可以做很多事情，而且性能很好。所以你真的想开始使用几何而不是点或形状的集合（它们更多 UI 实用程序）。

在这里，我简单地使用了几何学的combination特性，一个4行代码的算法：

public static IEnumerable<Rect> ComputeIntersectingSegments(Geometry geometry, double y, double width)
{
    // Add a geometry line to compute intersections.
    // A geometry must not be 0 thickness for combination to be meaningful.
    // So we widen the line by a very small size
    var line = new LineGeometry(new Point(0, y), new Point(width, y)).GetWidenedPathGeometry(new Pen(null, 0.01));

    // Intersect the line with input geometry and compute intersections
    var combined = Geometry.Combine(line, geometry, GeometryCombineMode.Intersect, null);
    foreach (var figure in combined.Figures)
    {
        // the resulting figure can be a complex thing
        // we just want the bounding box
        yield return new PathGeometry(new PathFigure[] { figure }).Bounds;
    }
}

public partial class MainWindow : Window
{
    public MainWindow()
    {
        InitializeComponent();

        // use a canvas to display shape and intersections
        var canvas = new Canvas();
        Content = canvas;

        // your polygon can be built as a geometry for example like this:
        // var myPolygon = Geometry.Parse("M50,50 L50,165 L140,165 L140,120 L70,120 L80,70 L140,70 L140,50");

        // build a 'o' shape for testing, add it to the canvas
        var circle1 = new EllipseGeometry(new Point(100, 100), 70, 70);
        var circle2 = new EllipseGeometry(new Point(100, 100), 40, 40);

        // exclude mode will compute the 'o' shape ...
        var oGeometry = new CombinedGeometry(GeometryCombineMode.Exclude, circle1, circle2);

        var oPath = new Path();
        oPath.Stroke = Brushes.Black;
        oPath.Fill = Brushes.LightSeaGreen;
        oPath.StrokeThickness = 2;
        oPath.Data = oGeometry;

        canvas.Children.Add(oPath);

        // test many heights
        for (int y = 0; y < Height; y += 25)
        {
            foreach (var segment in ComputeIntersectingSegments(oGeometry, y, Width))
            {
                // for our sample, we add each segment to the canvas
                // Height is irrelevant, we use 2 for tests
                var line = new Rectangle();
                Canvas.SetLeft(line, segment.X);
                Canvas.SetTop(line, segment.Y);
                line.Width = segment.Width;
                line.Height = 2;
                line.Stroke = Brushes.Red;
                line.StrokeThickness = 1;
                canvas.Children.Add(line);
            }
        }
    }
}

这是结果：

计算给定水平线处的形状宽度

Calculation of shape width at given horizontal line

c#

algorithm

math

geometry

line-intersection