The physics
What wets you is the rain's velocity relative to you.
vrain, you = vrain − vyou
Point the umbrella straight into that relative velocity. Say the rain falls straight down at vr. You walk at vm. Then tilt the umbrella forward by θ.
tan θ = vm / vr
A wind adds a sideways part to the rain. Walk with the wind at its speed, and the rain looks vertical.
Here "dry" means your head and shoulders stay in the umbrella's shadow. The dashed box shows them. The shadow runs along the rain as you see it.