SVG路径上的Catmull-Rom插值

时间：2015-06-10 06:05:19

标签： svg computational-geometry curve-fitting catmull-rom-curve

我正在尝试使用SVG路径创建高性能，美观的铅笔工具。

我正在记录鼠标坐标以绘制路径。为了获得高保真路径（精确到用户的动作），我需要为每个像素移动记录一个点。

保留路径中的每个点会产生大量的点 非常适合以后的协作功能（来回发送大量的点效率不高），以及解析每次我需要操纵它们时，巨大的路径都是瓶颈

在路径的线性区域上，删除冗余点，仅保留表示段所需的点 - 我使用Ramer-Douglas-Peucker算法执行此操作。

但是简化路径会将其转换为低保真多边形

此时路径实际上只是连接线 - 因此路径看起来是锯齿状的。

一种可能的解决方案是将路径点与Cubic Bezier连接 - 但是这在简化路径上不起作用。每个点之间的距离太大，以至于Cubic Bezier“坐得”不错，因此平滑的路径不再准确地表示用户的预期路径。

另一个解决方案是在原始路径上简单地使用“后处理”算法，例如Schneider's Algorithm - 这个算法实际上不会实时工作，因为它是一个性能损失

理想的解决方案

（我认为）可行的解决方案是使用Centripetal Catmull-Rom插值。

Centripetal Catmull Rom vs rest of Catmull-Rom variants

在我研究的所有算法中，这似乎是最有希望的：

  1. 它不会在狭窄的角落产生自我交叉
  2. 它更适合点，因此它更准确地代表了 原始路径。

    3. Catmull-Rom 是一种插入一系列算法的算法 常规x / y点或原始路径是否需要包含 曲线Δ

2 个答案:

答案 0 :(得分：11)

直接回答您的问题：

  1. 是。 Catmull-Rom样条曲线是一种插入一系列（x，y，z）点的算法。它将在每两个连续点之间生成三次多项式曲线。
  2. 您不能直接使用Catmull Rom样条曲线来获取SVG路径。您需要先将其转换为立方贝塞尔曲线。

    3. 对于由点P0，P1，P2和P3以及结序列t0，t1，t2，t3定义的曲线段，可以通过提供的递归公式计算向心Catmull-Rom样条（在点P1和P2之间定义）在https://en.wikipedia.org/wiki/Centripetal_Catmull%E2%80%93Rom_spline。因此，我在此不再赘述。

    要将其转换为三次贝塞尔曲线，您需要计算第一个 P1和P2的导数为

    M1 = (t2-t1)*(c1*(P1-P0)/(t1-t0) + c2*(P2-P1)/(t2-t1))
M2 = (t2-t1)*(d1*(P2-P1)/(t2-t1) + d2*(P3-P2)/(t3-t2))

    哪里

     c1 = (t2-t1)/(t2-t0),
 c2 = (t1-t0)/(t2-t0),
 d1 = (t3-t2)/(t3-t1),
 d2 = (t2-t1)/(t3-t1)

    然后你可以将它转换成具有4个控制点的三次贝塞尔曲线：Q0，Q1，Q2和Q3：

    Q0 = P1
Q1 = P1 + M1/3
Q2 = P2 - M2/3
Q3 = P2

答案 1 :(得分：3)

我使用插值立方体

  1. 上的示例点
    • 任意轴上的鼠标方向改变
    • 如果鼠标移动了一段距离以获得更高的精度可以添加：如果方向的角度变化太大（我不使用它）
    • 如果鼠标停止（如果你也使用鼠标上/下也可以忽略）

  2. 使用采样点作为插值三次

    的控制点

    见Interpolation cubic vs. Bezier cubic你发现插值与贝塞尔立方之间的转换

  3. 如果这还不够，您可以加入方向相同的行

    以这种方式举手绘图的例子：

    follow mouse

    SVG：

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</svg>

    如果您使用阈值，则大约为20KB，您可以显着降低点数。这是根据我的需要调整的。

    这里是鼠标在绘图过程中如何感觉（在绘图过程中实时捕捉）的示例

    spektre hand drawed

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