I never saw her without glasses before, so it threw me off when she got contact lenses.
If you throw off people who are chasing you or trying to find you, you do something unexpected that makes them unable to catch you or find you. He is said to have thrown off pursuers by pedaling across the Wisconsin state line He tried to throw police off the track of his lover
If something throws off a substance, it produces it and releases it into the air. The belt may make a squealing noise and throw off sooty black particles of rubber = give off
get rid of, free oneself from; mislead, misdirect; cause a diversion (used in combination, e.g. "The robber used a disguise to throw the policemen off his trail.")
If you throw off something that is restricting you or making you unhappy, you get rid of it. a country ready to throw off the shackles of its colonial past One day depression descended upon him, and wherever he went after that he could never throw it off. = cast off