>> Visual picture of my problem <<

In the attached picture I have this bezier-curve (or easing curve), represented by 4 coordinates:

P0 (0, 0)
P1 (0.7, 0)
P2 (0.3, 1)
P3 (1, 1)

Given a specific time that can range between zero and one, I need to find the corresponding value that intersect the bezier-curve.

In this example, when time is equal to 0.6, I was able to find out that the corresponding value is approximately 0.7702, since I was able to measure it on the software I'm using (Illustrator). What I need though is a mathematical formula to calculate the corresponding value given any time.

I have no idea how -__-


The $x$ equation is


For a given $x$, you need to solve this cubic equation for $t$. You can do it for example using the analytic formulas.


When you have $t$, plug it in the $y$ equation.

Not so simple.

  • $\begingroup$ Help me understand. I am assuming that "x" is the final value that I am looking for, right? If I am doing this right, when I plug in t = 0.6 it's pretty easy to do the calculation, but the equation returns x = 0.4528, instead of x = 0.7702 (approximately). Am I doing something wrong? $\endgroup$ – Francesco Aug 11 '18 at 16:24
  • $\begingroup$ @Francesco: no, you don't seem to have understood my post. From $x$ deduce $t$ and from $t$, $y$. I said not so simple. $\endgroup$ – Yves Daoust Aug 11 '18 at 16:54

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