Bezier curve(again): given Y, solve for X

[abeall]

In a http://en.wikipedia.org/wiki/B%C3%A9zier_curve" defined by p1, cp1, cp2, p2, is it possible to solve for X when Y is known?

I can't figure out how to determine if the bezier curve intersects Y. I can conclude that the bezier curve does intersect if p1 and p2 are either side of Y. I can conclude that the bezier curve does not intersect if p1, cp1, cp2, and p2 all fall on one side of Y, but in the case that p1 and p2 fall on one side, and one or both cp1 and cp2 fall on the other side of Y, I don't know, it may or may not intersect. Additionally, it may intersect 1, 2, or 3 times.

Beyond that, I don't know how to solve X when Y is know, or even if it is possible.

Thanks for any advice.

 

[hotvette]

I think the situation is really the same as in your other thread - need to solve a cubic. In a cubic bezier, you have:

x = f(t,C_x)...C_x are the control points
y = g(t,C_y)...C_y are the control points

For the given y, you can solve the 2nd equation for t (0-3 solutions possible), then calculate x from the determined t.

In special cases, parametric equations can be combined (e.g. equation of a circle), but in general, it isn't possible because the curves can loop. That's what makes parametric equations so powerful.

 

[tacman]

It is possible to solve for X when Y is known in a Bezier curve. Since a Bezier curve is a parametric curve, it means that the coordinates of each point on the curve can be expressed as a function of a single parameter, t. Therefore, to solve for X when Y is known, we need to find the value of t that corresponds to the given Y coordinate.

To do this, we can use the parametric equations for a Bezier curve, which are:

X = (1-t)^3*p1 + 3(1-t)^2*t*cp1 + 3(1-t)*t^2*cp2 + t^3*p2
Y = (1-t)^3*p1 + 3(1-t)^2*t*cp1 + 3(1-t)*t^2*cp2 + t^3*p2

We can rearrange these equations to solve for t:

t = (Y - (1-t)^3*p1 - 3(1-t)^2*t*cp1 - t^3*p2) / (3(1-t)*t^2*cp2)

Once we have the value of t, we can plug it back into the parametric equations to find the corresponding X coordinate. However, it's important to note that there may be multiple values of t that correspond to a given Y coordinate, meaning that the Bezier curve may intersect the Y coordinate multiple times. In this case, we would need to use the values of t to find the corresponding X coordinates for each intersection point.

In summary, it is possible to solve for X when Y is known in a Bezier curve, but it may require some additional calculations and consideration of multiple values of t. I hope this helps clarify the process for you.