Can someone describe to me the geometric intuition behind using a mapping $ ((x_1,y_1),(x_2,y_2)) \mapsto \frac{d_1(x_1,y_1)}{1+d_1(x_1,y_1)} + \frac{1}{2} \frac{d_2(x_2,y_2)}{1+d_2(x_2,y_2)} $ to define a metric on the product of the metric spaces $(X,d_1),(Y,d_2)$ ?
Of course I can check the axioms, but that doesn't give me any insight; so why was it defined like this and not differently (especially, why the $\frac{1}{2}$)? Why not use $ ((x_1,y_1),(x_2,y_2))$ $ \mapsto d_1(x_1,y_1) $ $+ d_2(x_2,y_2) $ ?