elliptic-equationsparametric
So I've got parametric ellipse equasion like in this post: What is the parametric equation of a rotated Ellipse (given the angle of rotation)
$$x(\alpha) = R_x \cos(\alpha) \cos(\theta) – R_y \sin(\alpha) \sin(\theta) + C_x \\
y(\alpha) = R_x \cos(\alpha) \sin(\theta) + R_y \sin(\alpha) \cos(\theta) + C_y$$
My question is how to find all $\alpha$ values for which ellipse intersects given vertical or horizontal line, e.g. $x=1$ or $y=2$.?
Try the following, where $t$ represents the main parameter, $\theta$ is the tilting angle as the sine wave wraps around the cylinder, $k$ is the frequency of the sine wave, and $a$ is the amplitude:
$$\begin{align*} x(t) &= \cos (t \cos \theta - a \sin \theta \sin (k t)) \\ y(t) &= \sin (t \cos \theta - a \sin \theta \sin (k t)), \\ z(t) &= t \sin \theta + a \cos \theta \sin (k t). \end{align*}$$
Fancy animated picture! 
The parametric formula of an Ellipse - at (0, 0) with the Major Axis parallel to X-Axis and Minor Axis parallel to Y-Axis:
$$ x(\alpha) = R_x \cos(\alpha) \\ y(\alpha) = R_y \sin(\alpha) $$
Where:
- $R_x$ is the major radius
- $R_y$ is the minor radius
To rotate any formula we use the rotation mapping:
$$
x = t \cos(\theta) - f(t) \sin(\theta) \\
y = t \sin(\theta) + f(t) \cos(\theta)
$$
Where:
- $\theta$ is the rotation angle
- $t$ is the parameter of the original function
- $f(t)$ is the original function
Once we put the Ellipse equation in the rotation equation we get: $$ x(\alpha) = R_x \cos(\alpha) \cos(\theta) - R_y \sin(\alpha) \sin(\theta) \\ y(\alpha) = R_x \cos(\alpha) \sin(\theta) + R_y \sin(\alpha) \cos(\theta) $$
To shift any equation from the center we add $C_x$ to the $x$ equation and $C_y$ to the $y$ equation.
Therefore the equation of a Rotated Ellipse is:
$$
x(\alpha) = R_x \cos(\alpha) \cos(\theta) - R_y \sin(\alpha) \sin(\theta) + C_x \\
y(\alpha) = R_x \cos(\alpha) \sin(\theta) + R_y \sin(\alpha) \cos(\theta) + C_y
$$
Where:
- $C_x$ is center X.
- $C_y$ is center Y.
- $R_x$ is the major radius.
- $R_y$ is the minor radius.
- $\alpha$ is the parameter, which ranges from 0 to 2π radians.
- $\theta$ is the Ellipse rotation angle.
Best Answer
You’ve got an equation of the form $A\cos\alpha+B\sin\alpha = C$, so you could solve it directly, getting an unpleasant-looking expression involving $\arcsin$ and $\arctan$. However, you can take advantage of the fact that you’re working with a scaled and translated ellipse for a potentially simpler solution.
The parameterization that you’ve got in your question can be viewed as the unit circle $(\cos\alpha,\sin\alpha)$ scaled, rotated and translated. So, if you undo this transformation, the problem is reduced to finding the intersection of a line with the unit circle. Specifically, the line $x=h$ back-maps to $(R_x\cos\theta)x-(R_y\sin\theta)y+C_x=h$, while the line $y=k$ back-maps to $(R_x\sin\theta)x+(R_y\cos\theta)y+C_y=k$.