Finding Y given X on a Cubic Bezier Curve?

Question

I need a method that allows me to find the Y-coordinate on a Cubic Bezier Curve, given an x-coordinate.

I've come across lots of places telling me to treat it as a cubic function then attempt to find the roots, which I understand. HOWEVER the equation for a Cubic Bezier curve is (for x-coords):

X(t) = (1-t)^3 * X0 + 3*(1-t)^2 * t * X1 + 3*(1-t) * t^2 * X2 + t^3 * X3

What confuses me is the addition of the (1-t) values. For instance, if I fill in the X values with some random numbers:

400 = (1-t)^3 * 100 + 3*(1-t)^2 * t * 600 + 3*(1-t) * t^2 * 800 + t^3 * 800

then rearrange it:

800t^3 + 3*(1-t)*800t^2 + 3*(1-t)^2*600t + (1-t)^3*100 -400 = 0

I still don't know the value of the (1-t) coefficients. How I am I supposed to solve the equation when (1-t) is still unknown?

Answer 1

There are three common ways of expressing a cubic bezier curve.

First x as a function of t

x(t) = sum( f_i(t) a_i )
     = (1-t)^3 * x0 + 3*(1-t)^2 * t * x1 + 3*(1-t) * t^2 * x2 + t^3 * x3

Secondly y as a function of x

y(x) = sum( f_i(x) a_i )
     = (1-x)^3 * y0 + 3*(1-x)^2 * x * y1 + 3*(1-x) * x^2 * y2 + x^3 * y3

These first two are mathematically the same, just using different names for the variables.

Judging by your description "find the Y-coordinate on a Cubic Bezier Curve, given an x-coordinate on it." I'm guessing that you've got a question using the second equation are are trying to rearrange the first equation to help you solve it, where as you should be using the second equation. If thats the case, then no rearranging or solving is required - just plug your x value in and you have the solution.

Its possible that you have an equation of the third kind case, which is the ugly and hard case. This is both the x and y parameters are cubic Beziers of a third variable t.

x(t) = sum( f_i(t) x_i )
y(t) = sum( f_i(t) y_i )

If this is your case. Let me know and I can detail what you need to do to solve it.