  Constructive techniques to establish state-independent uncertainty relations
for the sum of variances of arbitrary two observables are presented. We
investigate the range of simultaneously attainable pairs of variances, which
can be applied to a wide variety of problems including finding exact bound for
the sum of variances of two components of angular momentum operator for any
total angular momentum quantum number $j$ and detection of quantum
entanglement. Resulting uncertainty relations are state-independent,
semianalytical, bounded-error and can be made arbitrarily tight. The advocated
approach, based on the notion of joint numerical range of a number of
observables and uncertainty range, allows us to improve earlier numerical works
and to derive semianalytical tight bounds for the uncertainty relation for the
sum of variances expressed as roots of a polynomial of a single real variable.
