I have the following position function for a point:
$x(t) := C_x - (S_x-C_x) \cdot \cos(t\cdot\theta) + (S_y-C_y) \cdot \sin(t\cdot\theta) + t \cdot v_x$
$y(t) := C_y - (S_x-C_x) \cdot \sin(t\cdot\theta) - (S_y-C_y) \cdot \cos(t\cdot\theta) + t \cdot v_y$
These functions calculate the x and y coordinates for a point $S$ at time $0.0 \leq t \leq 1.0$, which is rotating around a center $C$ and moving by a vector $\overrightarrow{v}$.
I would now like to calculate the intersection of this movement with a static line described by $y := m \cdot x + c$
This means, which values for t satisfy the equation $y(t) = m \cdot x(t) + c$
Is there an algebraic solution for this (at least I couldn't find one because of the trigonometric functions)? Or does it only have a numerical solution?
In the latter case, the equation could be changed to:
$f(t) := y(t) - m \cdot x(t) - c$
The solutions are then all values t for which $f(t) = 0$
and be numerically solved with newtons method. The problem here is to find good start values and how to make sure, that all values have been found. Depending on the vector $v$ there may be an arbitrary number of solutions (but it should be possible to calculate the upper limit: $\left \lfloor{\frac{t \cdot \theta}{2\cdot\pi}}\right \rfloor + 2$).
Is there an error in my thoughts? Any suggestions on this?
Your setup is highly overloaded. What we see is a cycloid $\gamma$ and an extra line $\ell$. Kinematically $\gamma$ is the superposition of a circular motion having radius $R:=|S-C|$ and a forward motion in direction $\vec v$.
If $\ell$ is parallel to $\vec v$ the set $\gamma\cap\ell$ is either empty, or consists of infinitely many points which repeat periodically along the direction $\vec v$. This case should not be too difficult.
If $\ell$ is not parallel to $\vec v$ the line $\ell$ will intersect the trace $t\mapsto c(t)$ of the center of the circular motion in a point $O$. Choose this point as origin of your coordinate system, assume that the origin of time is chosen such that $c(0)=O$, and choose the $x$-axis in the direction of $\vec v$. If the rotation is counterclockwise the cycloid $\gamma$ then appears as $$\eqalign{x(t)&=R\cos(t+\alpha)+ v t\cr y(t)&=R\sin(t+\alpha)\cr}$$ with a certain $\alpha\in[0,2\pi[\ $, and $\ell$ assumes the form $y=mx$, see the figure below. The number of points $p\in\ell\cap\gamma$ is finite, and gets smaller when $|m|$ gets larger. Since all these points have a $y$-coordinate of absolute value $\leq R$ their $x$-coordinate is $\leq R/|m|$, and this bounds the $t$-range in which such points should be sought.