When working with parametric equations of a bezier curve, I know that when the image is rotated at an angle, you would use to find the new coordinates:
$\ \left(x\cos θ - y\sin θ, x\sin θ + y\cos θ\right) $
I have a bezier curve and when only$\ x $ is known I use a polynomial equation solver to solve for$\ t $. I then use that to find$\ y $.
Let's say I have on my curve when $\ t = .24 , x = .36, y = -.17 $
Then I rotate my curve at an angle of 30 degrees. I could use the rotation equation to find the new coordinates of both$\ x $ and$\ y $ at$\ t $, but that is not what I want to do. I want to find both$\ t $ and$\ y $ when$\ x $ is at the same value of $\ x=.36 $
I know I could find$\ t $ again using the new equation output by the rotation at 30 degrees and use the polynomial equation solver all over again to find$\ t $, and then$\ y $, but is there an easier way?
Is there some formula like there is for the$\ \left(x,y\right)$ coordinates for rotating the value of$\ t $ at an angle?