三次贝塞尔曲线-给定X可获得Y [英] Cubic bezier curves - get Y for given X

问题描述

我有三次贝塞尔曲线，其中给出了第一个点和最后一个点(即P0(0,0)和P3(1,1)). 其他两点的定义如下:cubic-bezier(0.25，0.1，0.25，1.0)(X1，Y1，X2，Y2，而且这些值分别不得小于或大于0或1)
现在，假设只有一个，我该怎么做才能获得给定X的Y坐标? (我知道在某些情况下可以有多个值，但我们将它们放在一边.我不是在这里进行火箭科学，我只是想每秒能够多次获得Y来进行转换)

I have a cubic bezier curve where the first and last points are given (namely P0(0,0) and P3(1,1)). The other two points are defined like this: cubic-bezier(0.25, 0.1, 0.25, 1.0) (X1, Y1, X2, Y2, also those values must not be smaller or larger than 0 or 1, respectively)
Now what would I have to do to get the Y coordinate for a given X, assuming there's only one? (I know that under certain circumstances there can be multiple values, but let's just put them aside. I'm not doing rocket science over here, I just want to be able to get Y multiple times per second to do transitions)

我设法对此进行了总结:给定x的y坐标立方贝塞尔曲线，但我不明白xTarget代表什么.
哦，这也不是什么功课，我对互联网上没有三次方贝塞尔曲线的可理解之处感到恼火.

I managed to dig up this: y coordinate for a given x cubic bezier, but I don't understand what xTarget stands for.
Oh, also this is no homework whatsoever, I'm just a bit annoyed at the fact that there's no comprehensible stuff about cubic bezier curves on the internet.

推荐答案

如果有

P0 = (X0,Y0)
P1 = (X1,Y1)
P2 = (X2,Y2)
P3 = (X3,Y3)

然后对于[0,1]中的任何t，您都会在坐标给出的曲线上得到一个点

Then for any t in [0,1] you get a point on the curve given by the coordinates

X(t) = (1-t)^3 * X0 + 3*(1-t)^2 * t * X1 + 3*(1-t) * t^2 * X2 + t^3 * X3
Y(t) = (1-t)^3 * Y0 + 3*(1-t)^2 * t * Y1 + 3*(1-t) * t^2 * Y2 + t^3 * Y3

如果为您提供了x值，则需要找到[0,1]中的哪些t值对应于曲线上的该点，然后使用这些t值来找到y坐标.

If you are given an x value, then you need to find which t values in [0,1] correspond to that point on the curve, then use those t values to find the y coordinate.

在上面的X(t)公式中，将左侧设置为x值，然后插入X0，X1，X2，X3.这为您提供了变量为t的三次多项式.您可以为t解决此问题，然后将该t值插入Y(t)公式以获得y坐标.

In the X(t) equation above, set the left side to your x value and plug in X0, X1, X2, X3. This leaves you with a cubic polynomial with variable t. You solve this for t, then plug that t value into the Y(t) equation to get the y coordinate.

解决三次多项式是棘手的，但是可以通过谨慎地使用其中一种方法来解决解决三次多项式.

Solving the cubic polynomial is tricky but can be done by carefully using one of the methods to solve a cubic polynomial.

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