  We provide two closed-form geodesic formulas for a family of metrics on
Stiefel manifold, parameterized by two positive numbers, having both the
embedded and canonical metrics as special cases. The closed-form formulas allow
us to compute geodesics by matrix exponential in reduced dimension for low-rank
manifolds. Combining with the use of Fr{\'e}chet derivatives to compute the
gradient of the square Frobenius distance between a geodesic ending point to a
given point on the manifold, we show the logarithm map and geodesic distance
between two endpoints on the manifold could be computed by {\it minimizing}
this square distance by a {\it trust-region} solver. This leads to a new
framework to compute the geodesic distance for manifolds with known geodesic
formula but no closed-form logarithm map. We show the approach works well for
Stiefel as well as flag manifolds. The logarithm map could be used to compute
the Riemannian center of mass for these manifolds equipped with the above
metrics. We also deduce simple trigonometric formulas for the Riemannian
exponential and logarithm maps on the Grassmann manifold.
