如何将三次曲线的两个控制点转换为二次曲线的单个控制点? [英] How do I convert the 2 control points of a cubic curve to the single control point of a quadratic curve?

问题描述

在网上搜索后，我在各个论坛中看到许多人都在暗示用二次曲线近似三次曲线.但是我找不到公式.

Having searched the web, I see various people in various forums alluding to approximating a cubic curve with a quadratic one. But I can't find the formula.

这是我想要的:

输入:startX，startY，control1X，control1Y，control2X，control2Y，endX，endY输出:startX，startY，controlX，controlY，endX，endY

input: startX, startY, control1X, control1Y, control2X, control2Y, endX, endY output: startX, startY, controlX, controlY, endX, endY

实际上，由于起点和终点是相同的，所以我真正需要的只是...

Actually, since the starting and ending points will be the same, all I really need is...

输入:startX，startY，control1X，control1Y，control2X，control2Y，endX，endY输出:controlX，controlY

input: startX, startY, control1X, control1Y, control2X, control2Y, endX, endY output: controlX, controlY

推荐答案

如上所述，从4个控制点到3个控制点通常是一个近似值.只有一种情况是精确的-三次贝塞尔曲线实际上是一个度数升高的二次贝塞尔曲线.

As mentioned, going from 4 control points to 3 is normally going to be an approximation. There's only one case where it will be exact - when the cubic bezier curve is actually a degree-elevated quadratic bezier curve.

您可以使用度数升高方程来得出近似值.很简单，结果通常很好.

You can use the degree elevation equations to come up with an approximation. It's simple, and the results are usually pretty good.

我们称三次Q0..Q3的控制点和二次P0..P2的控制点.然后，对于度数升高，公式为:

Let's call the control points of the cubic Q0..Q3 and the control points of the quadratic P0..P2. Then for degree elevation, the equations are:

Q0 = P0
Q1 = 1/3 P0 + 2/3 P1
Q2 = 2/3 P1 + 1/3 P2
Q3 = P2

在您的情况下，您有Q0..Q3，并且正在求解P0..P2.根据上面的公式，有两种方法可以计算P1:

In your case you have Q0..Q3 and you're solving for P0..P2. There are two ways to compute P1 from the equations above:

P1 = 3/2 Q1 - 1/2 Q0
P1 = 3/2 Q2 - 1/2 Q3

如果这是度高的三次方，则两个方程将为P1给出相同的答案.既然不太可能，那么最好的选择就是将它们平均化.因此，

If this is a degree-elevated cubic, then both equations will give the same answer for P1. Since it's likely not, your best bet is to average them. So,

P1 = -1/4 Q0 + 3/4 Q1 + 3/4 Q2 - 1/4 Q3

要翻译成您的条款，

controlX = -0.25*startX + .75*control1X + .75*control2X -0.25*endX

Y的计算方法类似-尺寸是独立的，因此适用于3d(或n-d).

Y is computed similarly - the dimensions are independent, so this works for 3d (or n-d).

这是一个近似值.如果需要更好的近似值，一种获得方法是通过使用deCastlejau算法细分初始三次方，然后对每个线段进行度缩减.如果您需要更好的连续性，则可以使用其他一些近似方法，这些方法不那么快捷和肮脏.

This will be an approximation. If you need a better approximation, one way to get it is by subdividing the initial cubic using the deCastlejau algorithm, and then degree-reduce each segment. If you need better continuity, there are other approximation methods that are less quick and dirty.

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