I'm following this friendly article in order to compute a biarc in $\mathbb{R}^2$. The author writes
Choosing a value for $d_1$ plays a big role in the shape of the biarc. Negative values will make it long and curved. Values near zero will make the first arc compact. $\textbf{Some values can even create a case without a solution}$.
The main ideia: we construct two circunferences that are tangent to each other, $C_1$ and $C_2$, from five inputs: a starting point $P_1$ and its tangent $\hat{t_1}$, an ending point $P_2$ and its tangent $\hat{t_2}$ and a scalar $d_1$ (please, see the diagram).
The author wrote the tangent point $\vec{p_m}=C_1\cap C_2$ as a linear combination of $d_1$ and $d_2$, before finding $d_2$: $$\vec{p_m}=(P_1+d_1\hat{t_1})\frac{d_2}{d_1+d_2}+(P_2-d_2\hat{t_2})\frac{d_1}{d_1+d_2}$$
He also wrote a nicer version of $\vec{p_m}$: $$\vec{p_m}=\vec{p_1}+c_1\hat{t_1}+\langle\vec{q}-d_1\hat{t_1},\hat{t_2}\rangle\hat{t_2}$$
Where $\vec{q}=P_2-P_1$
There are a couple of denominator indeterminations that are confusing me. The first one is for the scalar $d_2$: $$d_2=\frac{\langle\vec{q},\vec{q}\rangle-2d_1\langle\vec{q},\hat{t_1}\rangle}{2\langle\vec{q},\hat{t_2}\rangle-2d_1(\langle\hat{t_2},\hat{t_1}\rangle-1)}$$
Anoter two are the centers of both circunferences $C_1$ and $C_2$: $$C_1=\vec{p_1}+\frac{\langle\vec{p_m}-\vec{p_1},\vec{p_m}-\vec{p_1}\rangle}{2\langle\vec{p_m}-\vec{p_1},\hat{n_1}\rangle}\hat{n_1}\hspace{15mm} C_2=\vec{p_2}+\frac{\langle\vec{p_m}-\vec{p_2},\vec{p_m}-\vec{p_2}\rangle}{2\langle\vec{p_m}-\vec{p_2},\hat{n_2}\rangle}\hat{n_2}$$
Where $\hat{n_i}$ is the vector with $\langle\hat{n_i},\hat{t_i}\rangle=0$ at $P_i$. Also, the term multipliying $\hat{n_i}$ is the $C_i$ radius.
From the input values $P_1$, $P_2$, $\hat{t_1}$, $\hat{t_2}$ and $d_1$, I'd like to know which input value settings cause indetermination in the denominators along the construction of the biarc, $\textit{i.e.}$, the case(s) without a solution.
Ron pointed out in the comments that I need to better define when a biarc is invalid. I dont know exactly, but I think its fine to have zero radius because it could(?) produce straight segments connecting the points $P_1$ and $P_2$. The possibility of an infinite radius would be my second question, because I think it might lead to a line that overshoots the points when the circunference's curvature $\kappa=1/r$ changes sign.
After following the article, I wrote the procedure described in it to produce this animation (just press play).
Thank you for your time.