I want to compute intersection point of two curves given by two parametric equations as follows:
$$x_{1}(t_{1})=r_{1}⋅\sin(t_{1})−r_{1}⋅t_{1}⋅\cos(t_{1})$$ $$y_{1}(t_{1})=r_{1}⋅\cos(t_{1})+r_{1}⋅t_{1}⋅\sin(t_{1})$$
and
$$x_{2}(t_{2})=r_{2}⋅\cos(t_{2})$$ $$y_{2}(t_{2})=r_{2}⋅\sin(t_{2})$$
The first one is involute equation and the second one is circle eq. My question is if I can do this using bisection method? I am able to solve this using Newtons method but as I am developing something a little bit more complex, the Newtons one not always converges so I want to combine these two methods roughly computing initial values of $t_{1}$ and $t_{2}$ using bisection and then using Newtons one. The problem comes to solving:
$$x(t_{1})-x(t_{2})=0$$ $$y(t_{1})-y(t_{2})=0$$
Secondly what I am asking for is some hints of how I need to rewrite conditions of bisection algorithm to include the fact that these curves are represented in parametric form? I am attaching the picture of the plot of these curves withing given intervals of $t_{1}$ and $t_{2}$, where the black one is circle and the red one is involute ($r_{1}<r_{2}$).
Yes. It appears that you have a complicated problem consisting of two equations in two variables. But appearances are deceiving and there is a short-cut. One curve is parameterized by $$\gamma_1(t) = (x_1(t),y_1(t)).$$ The second curve (the circle) is a level curve for the function $f : \mathbb{R}^2 \rightarrow \mathbb{R}$ given by $$f(x,y) = x^2 + y^2.$$ Specifically, the level curve given by $$\Gamma_2 = \{ (x,y) \: : \: f(x,y) = r_2^2\}.$$ The two curves intersect when $$f(\gamma_1(t)) = r_2^2.$$ You can apply the bisection method to the function $$g(t) = f(\gamma_1(t)) - r_2^2,$$ provided that you can locate points $a$ and $b$ such that $g(a)$ and $g(b)$ have different signs. Finding such a bracket around a root is not necessarily trivial and trial and error is not the worst approach. Once an intersection point $\gamma(t_1)$ has been determined, you can immediately solve for $t_2$.
It is clear that the standard bisection method can be applied when one curve is a level curve for a function which is easy to identify. The exists multivariate bisection methods which apply to systems of equations of multiple variables but they are not needed here.
Implementing the standard bisection method correctly and robustly is a subject for another question because it involves bounding the rounding error.