Someone who stays calm and unemotional is unblinking. The unblinking gaze of your social studies teacher may be enough to quiet the entire class.
This adjective is also useful for eyes that literally don't blink: "The lizard's unblinking eyes made me nervous, so I covered the cage." You can also use it to mean "brave or unflinching," like an unblinking warrior who marches into battle with her sword raised. This word is composed of the prefix un- ("not" or "opposite of") and the verb blink, which may come from the Middle Dutch blinken, "to glitter."