三次方贝塞尔曲线笔触轮廓 [英] Outline of cubic bezier curve stroke

问题描述

通过三次贝塞尔曲线的笔触，我的意思是绘制具有特定线宽"w"的曲线"A".

By stroke of the cubic bezier curve I mean rendering a curve 'A' with a specific line width 'w'.

如何导出其他描述贝塞尔曲线'A'的轮廓的三次贝塞尔曲线?

How can I derive other cubic bezier curves that describe the outline of the stroke of bezier 'A' ?

推荐答案

哦.您想要获得贝塞尔曲线的偏移曲线.

Ohhh. You want to get the offset-curves of an bezier curve.

坏消息.这很难，因为这些曲线不能简单地由数值导出.它们包含各种交集，循环和其他令人讨厌的东西.

Bad news. this is hard because these curves can't be simply derived numerical. They contain all kinds of intersections, loops and other nasty stuff.

尽管有一些近似值.到目前为止，我读过的最好的方法来自Thomas F. Hain(

There are some approximations though. The best approach I've read so far is from a paper by Thomas F. Hain (Fast, Precise Flattening of Cubic Bézier Path and Offset Curves).

他确实变平，所以他的论文主要是关于将偏移曲线分解为线段和圆弧段，但是您可以稍后将它们合并回贝塞尔曲线.

He does flattening, so his paper is mostly about decomposes the offset curves into line-segments and circular arc-segments, but you can merge them back to beziers later.

为了更好地理解，您可能还需要阅读他与bezier相关的其他文章.

For better understanding you may want to read his other bezier related papers as well.

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