在SVG路徑上的Catmull-Rom插值




computational-geometry curve-fitting (2)

直接回答你的問題:

  1. 是。 Catmull-Rom樣條是一個插值一系列(x,y,z)點的算法。 它將在每兩個連續的點之間生成一個三次多項式曲線。
  2. 您不能直接使用Catmull Rom樣條來獲取SVG路徑。 您需要先將其轉換為三次貝塞爾曲線。

對於由點P0,P1,P2和P3以及結點序列t0,t1,t2,t3定義的曲線段,向心Catmull-Rom樣條(定義在點P1和P2之間)可以通過https: //en.wikipedia.org/wiki/Centripetal_Catmull%E2%80%93Rom_spline 。 所以我不在這裡詳細說明。

要將其轉換為三次貝塞爾曲線,需要計算P1和P2的一階導數為

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))

哪裡

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

然後你可以把它轉換成四個控制點的三次貝塞爾曲線:Q0,Q1,Q2和Q3:

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

我正在嘗試使用SVG路徑創建高性能,美觀的鉛筆工具。

我正在記錄鼠標坐標繪製一條路徑。 為了獲得高保真的路徑(精確到用戶的運動),我需要為每個像素運動記錄一個點。

保持路徑中的每一個點都會產生大量的點,這對後來的協作功能來說並不理想(來回發送大量的點不是很有效),而且每次我們需要操縱它們時都要分析巨大的路徑一個瓶頸

在路徑的線性區域上,刪除冗餘點,只保留表示分段的點 - 我使用Ramer-Douglas-Peucker算法進行此操作。

但是,簡化路徑會變成低保真的多邊形

在這一點上,路徑實際上只是連接線 - 因此路徑看起來鋸齒狀。

一個可能的解決方案是連接路徑點與Cubic Bezier的 - 但是這在簡化路徑上不起作用。 每個點之間的距離對Cubic Bezier來說“太好”,因此平滑的路徑不再準確地表示用戶的預期路徑。

另一種解決方案是在原始路徑上簡單地使用“後處理”算法,例如施耐德算法( Schneider's Algorithm) - 該算法實際上並不實時工作,但是因為它是性能豬

理想的解決方案

(我認為)可以工作的解決方案是使用Centripetal Catmull-Rom插值。

在我研究的所有算法中,這似乎是最有希望的:

  1. 它不會在狹窄的角落形成自交
  2. 它更貼合點,因此它更準確地代表原來的路徑。

Catmull-Rom是一種插值一系列常規x / y點的算法還是原始路徑需要由曲線組成?


我用這個插值立方體

  1. 採樣點

    • 鼠標方向改變在任何軸
    • 如果鼠標走了一段距離以獲得更高的精度,可以加上:如果方向角度變化太大(我不用這個)
    • 如果鼠標停止(如果您使用鼠標上/下也可以忽略)
  2. 使用採樣點作為插值立方體的控制點

    見插值立方與貝塞爾立方,你也發現插值和貝塞爾立方之間的轉換

  3. 如果這還不夠,你可以加入同一方向的線

    這裡有一個手繪的例子:

    而SVG:

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    </svg>

    如果使用閾值,則大約20KB,可以顯著降低點數。 這是調整,以適應我的需求。

    下面是繪製過程中鼠標如何跟隨感覺的示例(在繪圖過程中實時捕獲)





catmull-rom-curve