Let $x = \sum_{i=1}^N a_i \cos^2(\theta_i)$ and $y = \sum_{i=1}^N a_i \cos^3(\theta_i)$ be the two convex combinations of powers of cosines, where the variables satisfy the following constraints
(1). $\sum_{i=1}^N a_i = 1$;
(2). $\forall i\,,\, 1\ge a_i\ge 0$;
(3). $\sum_{i=1}^N a_i \cos(\theta_i) = 0$.
(The cosine function can be replaced with the constraint that these variables are in $[-1,1]$.)
What is the feasible region $\mathcal{S}$ for $(x, y)$? Is there a closed-form expression of the boundary that depends on $N$?
What I have tried:
If $(x,y)\in \mathcal{S}$, then $(x,-y)\in \mathcal{S}$.
When $N = 1$, $x = \cos^2(\theta)$, $y = \cos^3(\theta)$, and condition (3) becomes $\cos(\theta) = 0$.Thus, the feasible region is a single point $(0,0)$.
When $N = 2$, $x = a\cos^2(\theta_1) + (1-a)\cos^2(\theta_2)$, $y = a\cos^3(\theta_1) + (1-a)\cos^3(\theta_2)$, and condition (3) becomes $a\cos(\theta_1) = (a-1)\cos(\theta_2)$, $1\ge a\ge 0$.
If $a = 0$, then $(x,y)=(0,0)$.
If $a>0$, then $x = (1-a)/a\cos^2(\theta_2)$, $y = (2a-1)(1-a)/a^2\cos^3(\theta_2)$. The feasible region is given by the following plot:
It seems that the feasible region $\mathcal{S}$ is convex, that is for all $(x_1,y_1)\in\mathcal{S}$ and $(x_2,y_2)\in\mathcal{S}$, there is $\lambda (x_1,y_1) + (1-\lambda)(x_2,y_2)\in\mathcal{S}$ for $1\ge \lambda\ge 0$.
Thanks for all the comments and suggestions.