algorithm - cornu螺旋/样条的高速绘图算法？

Question

有没有人见过角形螺旋线(也称为回旋曲线)或样条曲线的合适绘图算法？对于圆弧和直线，我们有 Bresenham 算法之类的东西。这适用于回旋曲线吗？​​

维基百科页面有这个 Sage 代码:

p = integral(taylor(cos(L^2), L, 0, 12), L)
q = integral(taylor(sin(L^2), L, 0, 12), L)
r1 = parametric_plot([p, q], (L, 0, 1), color = 'red')

是否有可用的参数图示例代码？我的网络搜索结果不多。

Answer 1

我没有看到任何现有的高速算法。但是，我确实了解了绘制这种东西的常用方法。本质上，您递归地拆分 L，直到计算出的左、中和右点足够接近一条直线，您可以绘制该直线。我能够使用 MathNet.Numerics.dll 进行集成。一些代码:

    public static void DrawClothoidAtOrigin(List<Point> lineEndpoints, Point left, Point right, double a, double lengthToMidpoint, double offsetToMidpoint = 0.0)
    {
        // the start point and end point are passed in; calculate the midpoint
        // then, if we're close enough to a straight line, add that line (aka, the right end point) to the list
        // otherwise split left and right

        var midpoint = new Point(a * C(lengthToMidpoint + offsetToMidpoint), a * S(lengthToMidpoint + offsetToMidpoint));
        var nearest = NearestPointOnLine(left, right, midpoint, false);
        if (Distance(midpoint, nearest) < 0.4)
        {
            lineEndpoints.Add(right);
            return;
        }
        DrawClothoidAtOrigin(lineEndpoints, left, midpoint, a, lengthToMidpoint * 0.5, offsetToMidpoint);
        DrawClothoidAtOrigin(lineEndpoints, midpoint, right, a, lengthToMidpoint * 0.5, offsetToMidpoint + lengthToMidpoint);
    }

    private static double Distance(Point a, Point b)
    {
        var x = a.X - b.X;
        var y = a.Y - b.Y;
        return Math.Sqrt(x * x + y * y);
    }

    private static readonly double PI_N2 = Math.Pow(Math.PI * 2.0, -0.5);

    public static double C(double theta)
    {
        return Integrate.OnClosedInterval(d => Math.Cos(d) / Math.Sqrt(d), 0.0, theta) / PI_N2;
    }

    public static double S(double theta)
    {
        return Integrate.OnClosedInterval(d => Math.Sin(d) / Math.Sqrt(d), 0.0, theta) / PI_N2;
    }