给定具有固定端点的三次方贝塞尔曲线，当给定y位置进行检查时，如何找到沿该点的x位置? [英] Given a cubic Bezier curve with fixed endpoints, how can I find the x position of a point along it when given a y position to check?

问题描述

假设我有一个Bezier曲线，其中有两个固定端点，一个端点位于 x(0)，y(1)，另一个端点位于 x(1)，y(0)(左下角和右上角)现在，我们有两个控制点，它们可以位于x(0)，x(1)，y(0)和y(1)之间的任何位置.对于这个问题，我只想说控制点#1在x(0.1)y(0.6)处，而控制点#2在x(0.9)和y(0.4)处.(这假设一个从左上方"坐标系)

Let's say I have a Bezier curve with two fixed endpoints, one at x(0), y(1) and one at x(1), y(0) (bottom left corner and upper right corner) Now let's say I have two control points, which can be at any locations between x(0), x(1), y(0), and y(1). For this question, I'll just say that control point #1 is at x(0.1) y(0.6) and control point #2 is at x(0.9) and at y(0.4). (This assumes a "from upper left" coordinate system)

这是我们曲线的一个小例子:

Here's a small illustration of our curve:

现在假设我的y位置为0.7.该数学公式看起来如何确定对应的x位置到y(0.7)的点是什么?我该怎么办?

Now let's say I'm given a y position of 0.7. What would the math look like to figure out what the corresponding x position is to the point at y(0.7)? How would I do this?

抱歉，这个问题不属于这里，但我认为这是编码中经常遇到的问题，很可能你们中的许多人都在寻找我想要的答案.

Sorry if this question doesn't belong here, but I figured this is a common problem faced in coding and that it's likely that many of you have the answer I'm looking for.

推荐答案

您具有函数 X(t)和 Y(t)的三次方程，其中t 是曲线参数(曲线上点的范围为 0..1 ).在伯恩斯坦多项式基础上(曲线定义的常用形式):

You have cubic equation for functions X(t) and Y(t) where t is curve parameter (range 0..1 for points on curve). In Bernstein polynomial basis (usual form for curve definition):

X(t) = P0.X*(1-t)^3+3*P1.X*(1-t)^2*t+3*P2.X*(1-t)*t^2+P3.X*t^3
Y(t) = P0.Y*(1-t)^3+3*P1.Y*(1-t)^2*t+3*P2.Y*(1-t)*t^2+P3.Y*t^3

具有 Y 值，我们可以找到相应的 t 参数-请注意，在 0..1 .Y分量的幂表示:

Having Y value, we can find corresponding t parameters - note there might be from 0 to 3 possible roots in range 0..1. Representation of Y-component in power basis:

Y(t) = P0.Y*(1-t)^3+3*P1.Y*(1-t)^2*t+3*P2.Y*(1-t)*t^2+P3.Y*t^3 = 
       t^3*(P3Y-3P2Y+3P1Y-P0Y) + t^2*(3P2Y-6P1Y+3P0Y) + t^2*(3P1Y-3P0Y) + (P0Y) = 
       t^3*a + t^2*b + t^2*c + d' = y_position 

最后的三次方程是:

t^3*a + t^2*b + t^2*c + d = 0
   where
a = P3.Y-3*P2.Y+3*P1.Y-P0.Y
b = 3*P2.Y-6*P1.Y+3*P0.Y
c = 3*P1.Y-3*P0.Y
d = P0.Y - y_position

 

求解三次方程，以计算 t (也许有些值用于波浪曲线)

Solve cubic equation to calculate t (perhaps some values for wavy curves)

然后为给定的 t 计算相应的 X 值:

Then for given t calculate corresponding X value:

X(t) = P0.X*(1-t)^3+3*P1.X*(1-t)^2*t+3*P2.X*(1-t)*t^2+P3.X*t^3

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