Uniform Point in the Unit Disk Without Rejection

You have access to an i.i.d. source of uniform random variables $U_1, U_2 \sim \text{Unif}(0,1)$. Construct a function $(X, Y) = f(U_1, U_2)$ such that $(X, Y)$ is uniformly distributed over the unit disk $\{(x, y) \in \mathbb{R}^2 : x^2 + y^2 \le 1\}$, without using rejection sampling. Prove that your construction produces a uniform distribution by deriving the joint density (or using a measure-preserving argument). Explicitly state any coordinate transformations you use.