Consider $b_i=(\theta_p-\theta_q)\sin((\theta_p-\theta_q)\alpha)$ where $p,q\in\{1,\cdots,w\}$ and $N=\frac{w(w-1)}{2}$. In order to guarantee the one-to-one correspondence between $(\theta,\alpha)$ and $b_i$, we restrict $(\theta,\alpha)\in [-\pi/2,\pi/2]^w\times (0,1]$. Then we can think of $b_i$ as a function $f(\theta,\alpha)$.
Denote $B:=(b_1,\cdots,b_N)$. Now let $B$ belongs to a subspace $\Phi\subset\mathbb{R}^N$ whose basis are $B_1,\cdots,B_m$ ($m$ denotes the dimension $\Phi$). Then for each $B\in\Phi$ we can find a $(\theta,\alpha)$ such that $f(\theta,\alpha)=B$, or say $(\theta,\alpha)=f^{-1}(B)$.
My question is, whether the set of $\{(\theta,\alpha)|f(\theta,\alpha)=B, \forall B\in\Phi\}$ is a connected set in $\mathbb{R}^{w+1}? if not, what conditions we could add in order to guarantee this?
I appreciate for any comment, suggestion!