SVG 路径上的 Catmull-Rom 插值
Catmull-Rom interpolation on SVG Paths
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），我需要为每个像素运动记录一个点。
保留路径中的每个点都会产生大量点，这不是以后协作功能的理想选择（来回发送大量点不是高效），再加上每次我需要操作它们时解析巨大的路径是一个瓶颈
在路径的线性区域，删除了冗余点，仅保留表示该线段所需的点 - 我使用 Ramer-Douglas-Peucker 算法执行此操作。
但简化路径会将其变成低保真多边形
此时路径实际上只是连接的线 - 因此路径看起来参差不齐。
一种可能的解决方案是将路径点与三次贝塞尔曲线连接起来——但这在简化路径上效果不佳。每个点之间的距离对于三次贝塞尔曲线来说太大 "sit" 很好，因此平滑的路径不再准确地代表用户的预期路径。
另一种解决方案是在原始路径上简单地使用 "post-processing" 算法，例如 Schneider's Algorithm - 这个算法实际上不会实时工作，因为它是一个性能猪
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。
在我研究的所有算法中，这似乎是最有希望的，因为：
- 它不会在急转弯处创建自相交
- 它更贴合点，因此更准确地代表了
原路径.
是Catmull-Rom一种算法
常规 x/y 点或原始路径是否需要由
曲线？
我为此使用插值三次
样本点在
- 任意轴鼠标方向变化
- 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（我不使用这个）
- 如果鼠标停止（如果你也用鼠标 up/down 可以忽略）
将采样点作为控制点进行三次插值
看Interpolation cubic vs. Bezier cubic你会发现插值和贝塞尔三次之间的转换
如果这还不够你可以加入相同方向的线
这里有这样手绘的例子：
还有 SVG：
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</svg>
它大约是 20KB，如果你玩阈值，你可以显着降低点数。这已根据我的需要进行了调整。
这里是绘图过程中鼠标跟随的感觉示例（绘图过程中实时捕获）
直接回答您的问题：
- 是的。 Catmull-Rom 样条是一种对一系列 (x, y, z) 点进行插值的算法。它将在每两个连续点之间生成一条三次多项式曲线。
- 您不能直接将 Catmull Rom 样条用于 SVG 路径。您需要先将其转换为三次贝塞尔曲线。
对于由点 P0、P1、P2 和 P3 以及节点序列 t0、t1、t2、t3 定义的曲线段，向心 Catmull-Rom 样条（定义在点 P1 和 P2 之间）可以通过递归计算https://en.wikipedia.org/wiki/Centripetal_Catmull%E2%80%93Rom_spline 中提供的公式。因此，我不会在这里详细说明。
要将其转换为三次贝塞尔曲线，您需要首先计算
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为具有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
On the Parameterization of Catmull-Rom Curves
见公式[2]
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），我需要为每个像素运动记录一个点。
保留路径中的每个点都会产生大量点，这不是以后协作功能的理想选择（来回发送大量点不是高效），再加上每次我需要操作它们时解析巨大的路径是一个瓶颈
在路径的线性区域，删除了冗余点，仅保留表示该线段所需的点 - 我使用 Ramer-Douglas-Peucker 算法执行此操作。
但简化路径会将其变成低保真多边形
此时路径实际上只是连接的线 - 因此路径看起来参差不齐。
一种可能的解决方案是将路径点与三次贝塞尔曲线连接起来——但这在简化路径上效果不佳。每个点之间的距离对于三次贝塞尔曲线来说太大 "sit" 很好，因此平滑的路径不再准确地代表用户的预期路径。
另一种解决方案是在原始路径上简单地使用 "post-processing" 算法，例如 Schneider's Algorithm - 这个算法实际上不会实时工作，因为它是一个性能猪
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。
在我研究的所有算法中，这似乎是最有希望的，因为：
- 它不会在急转弯处创建自相交
- 它更贴合点，因此更准确地代表了 原路径.
是Catmull-Rom一种算法 常规 x/y 点或原始路径是否需要由 曲线？
我为此使用插值三次
样本点在
- 任意轴鼠标方向变化
- 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（我不使用这个）
- 如果鼠标停止（如果你也用鼠标 up/down 可以忽略）
将采样点作为控制点进行三次插值
看Interpolation cubic vs. Bezier cubic你会发现插值和贝塞尔三次之间的转换
如果这还不够你可以加入相同方向的线
这里有这样手绘的例子：
还有 SVG：
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0.423589 -0.282842 3.600513 -2.899132 3.812308 -3.181974 c 0.423589 -0.353552 1.623761 -2.333447 1.906154 -2.545579 c 0.211794 -0.282842 2.400342 -1.555631 2.541539 -1.909184 c 0.282393 -0.212131 0.917777 -2.828421 1.270769 -3.181974 c 0.141196 -0.353552 3.035727 -2.757711 3.176923 -3.181974 c 0.352991 -0.353552 0.988376 -3.889079 1.270769 -3.818369 c 0.141196 -0.424263 2.47094 0.282842 2.541539 0.636394 c 0.282393 0.07071 0.635384 2.757711 0.635384 3.181974 c 0.070598 0.353552 -0.070598 3.464816 0 3.818369 c 0 0.424263 0.564786 2.61629 0.635384 3.181974 c 0.070598 0.353552 0.564786 4.666895 0.635384 5.091159 c 0.070598 0.565684 0.494188 3.464816 0.635384 3.818369 c 0.070598 0.424263 0.988376 2.757711 1.270769 3.181974 c 0.141196 0.353552 1.976752 3.818369 2.541539 3.818369 c 0.282393 0.424263 4.800684 0 5.083078 0 c 0.564786 0 2.04735 0 2.541539 0 c 0.282393 0 4.094701 0 4.447693 0 c 0.494188 0 2.753333 0 3.176923 0 c 0.352991 0 3.388718 0.212131 3.812308 0 c 0.423589 0 3.459317 -1.697053 3.812308 -1.909184 c 0.423589 -0.212131 2.823932 -1.555631 3.176923 -1.909184 c 0.352991 -0.212131 2.89453 -2.969842 3.176923 -3.181974 c 0.352991 -0.353552 2.259145 -1.555631 2.541539 -1.909184 c 0.282393 -0.212131 2.259145 -2.899132 2.541539 -3.181974 c 0.282393 -0.353552 2.188547 -2.192026 2.541539 -2.545579 c 0.282393 -0.282842 2.89453 -2.828421 3.176923 -3.181974 c 0.352991 -0.353552 2.47094 -2.899132 2.541539 -3.181974 c 0.282393 -0.353552 0.635384 -2.545579 0.635384 -2.545579 c 0.070598 -0.282842 0 0 0 0 "/> </svg>它大约是 20KB，如果你玩阈值，你可以显着降低点数。这已根据我的需要进行了调整。
这里是绘图过程中鼠标跟随的感觉示例（绘图过程中实时捕获）
直接回答您的问题：
- 是的。 Catmull-Rom 样条是一种对一系列 (x, y, z) 点进行插值的算法。它将在每两个连续点之间生成一条三次多项式曲线。
- 您不能直接将 Catmull Rom 样条用于 SVG 路径。您需要先将其转换为三次贝塞尔曲线。
对于由点 P0、P1、P2 和 P3 以及节点序列 t0、t1、t2、t3 定义的曲线段，向心 Catmull-Rom 样条（定义在点 P1 和 P2 之间）可以通过递归计算https://en.wikipedia.org/wiki/Centripetal_Catmull%E2%80%93Rom_spline 中提供的公式。因此，我不会在这里详细说明。
要将其转换为三次贝塞尔曲线，您需要首先计算 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为具有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
On the Parameterization of Catmull-Rom Curves 见公式[2]