Finding joint distribution function?

Question

Let $U$ and $V$ be two independent uniform $(0,1)$ random variables and let \begin{eqnarray*} R &=&\sqrt{\frac{U^{2}+V^{2}}{2}},\\ H &=&\frac{2U}{U+V}. \end{eqnarray*} I found the marginal cumulative distribution function (cdf) of $R$ and $H$. The cumulative distribution functions (cdf) of $R$ and $H$ are given by \begin{equation} F_{R}(t)=\left\{ \begin{array}{llll} 0,&~~ \mbox{ $t<0$},\\ \frac{\pi t^2}{2}, &~~ \mbox{ $0\leq t<\frac{1}{\sqrt{2}}$},\\ t^2\left(\frac{\pi}{2}-2\arccos(\frac{1}{t\sqrt{2}})\right)+\sqrt{2t^2-1}, & ~~ \mbox{ $\frac{1}{\sqrt{2}}\leq t< 1$},\\ 1,&~~ \mbox{ $t\geq 1$},\\ \end{array} \right. \end{equation} and \begin{equation} F_{H}(t)=\left\{ \begin{array}{lll} 0,&~~ \mbox{ $t<0$},\\ t+\frac{t^2}{2}\ln(\frac{2-t}{t}), &~~ \mbox{ $0\leq t<1$},\\ 1,&~~ \mbox{ $t\geq 1$},\\ \end{array} \right. \end{equation} respectively. I want to find the joint cdf of $R$ and $H$, i.e., $$F_{R,H}(r,h)= Pr(R‎\leq r, H\leq h).$$